The case q=3 and nu=0 : guessing the three unknown series M(1), M'(1) and M"(1)   (Section 13.1, continued)

>    subs(w=1,q=3,nu=0,eqM);

1+x*y*t*(2*y+1)*M(x,y)*M(1,y)-x*y*t*M(x,y)*M(x,1)-x*y*t*(x*M(x,y)-M(1,y))/(x-1)+x*y*t*(y*M(x,y)-M(x,1))/(y-1)-M(x,y)

The coefficient of t^n in M(x,y)

>    co:=proc(n) option remember:
if n=0 then 1
else
normal(x*y*(2*y+1)*add(co(i)*subs(x=1,co(n-1-i)),i=0..n-1)
-x*y/(x-1)*(x*co(n-1)-subs(x=1,co(n-1)))
-x*y*add(co(i)*subs(y=1,co(n-1-i)),i=0..n-1)
+x*y/(y-1)*(y*co(n-1)-subs(y=1,co(n-1)))): fi: end:

Let us guess whta is  M(1)

>   

>    Liste:=[seq(subs(x=1,y=1,co(n)),n=0..40)];

Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...
Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...
Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...
Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...
Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...
Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...
Liste := [1, 2, 10, 68, 544, 4816, 45700, 456168, 4732480, 50612672, 554769760, 6205681664, 70611988032, 815235991680, 9530929051872, 112650592638400, 1344326707716096, 16179839817783040, 1962201150619...

>    with(gfun):

>    ser:=listtoseries(Liste,t);

ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...
ser := series(1+2*t+10*t^2+68*t^3+544*t^4+4816*t^5+45700*t^6+456168*t^7+4732480*t^8+50612672*t^9+554769760*t^10+6205681664*t^11+70611988032*t^12+815235991680*t^13+9530929051872*t^14+112650592638400*t^1...

>    with(numapprox);

[chebdeg, chebmult, chebpade, chebsort, chebyshev, confracform, hermite_pade, hornerform, infnorm, laurent, minimax, pade, remez]

>    L:=[1,ser,ser^2,ser^3,ser^4]:

>    hermite_pade(L,t,35);

[1-40*t+540*t^2-2720*t^3+432*t^4, -1+42*t-536*t^2+1712*t^3+9040*t^4-864*t^5, -78*t^2+2040*t^3-14432*t^4-7200*t^5, 24000*t^5-1704*t^4, -12500*t^6]

Conjectured equation for M(1)

>    algM1:=collect(1-2720*t^3-40*t+432*t^4+540*t^2+ M1*(-1+1712*t^3+42*t+9040*t^4-536*t^2-864*t^5)+
M1^2*(2040*t^3-14432*t^4-78*t^2-7200*t^5)+M1^3*(24000*t^5-1704*t^4)+M1^4*(-12500*t^6),M1,factor);

algM1 := -12500*M1^4*t^6+24*t^4*(1000*t-71)*M1^3-2*t^2*(39-1020*t+7216*t^2+3600*t^3)*M1^2+M1*(-1+42*t-536*t^2+1712*t^3+9040*t^4-864*t^5)+1-40*t+540*t^2-2720*t^3+432*t^4
algM1 := -12500*M1^4*t^6+24*t^4*(1000*t-71)*M1^3-2*t^2*(39-1020*t+7216*t^2+3600*t^3)*M1^2+M1*(-1+42*t-536*t^2+1712*t^3+9040*t^4-864*t^5)+1-40*t+540*t^2-2720*t^3+432*t^4

>    degree(algM1,t);

6

>    genus(algM1,t,M1);

0

>    factor(parametrization(algM1,t,M1,S));

[(S^3-6*S^2+12*S-10)/S^3/(S-2), S*(S-2)*(S^4-8*S^3+22*S^2-28*S+12)/(S^3-6*S^2+12*S-10)^2]

>    tS:=op(1,%);

tS := (S^3-6*S^2+12*S-10)/S^3/(S-2)

Changing the parametrization so that it has a nicer form: the new parametrization is the one that appears in Theorem 25

>    tS:=factor(subs(S=2+S,S=1/S,tS));

tS := -(-1+2*S^3)*S/(1+2*S)^3

>    M1Ssol:=factor(solve(op(2,factor(subs(t=tS,algM1))),M1));

M1Ssol := -(1+2*S)*(4*S^4+4*S^3+2*S^2-1)/(-1+2*S^3)^2

>    algS:=numer(t-tS);

algS := t+6*t*S+12*t*S^2+8*S^3*t-S+2*S^4

>    Sser:=op(2,algeqtoseries(algS,t,S,15,true));

Sser := series(1*t+6*t^2+48*t^3+442*t^4+4428*t^5+46920*t^6+517424*t^7+5878560*t^8+68343600*t^9+809244016*t^10+9725962656*t^11+118345000128*t^12+1455070212096*t^13+18049788630144*t^14+225624841265664*t^...
Sser := series(1*t+6*t^2+48*t^3+442*t^4+4428*t^5+46920*t^6+517424*t^7+5878560*t^8+68343600*t^9+809244016*t^10+9725962656*t^11+118345000128*t^12+1455070212096*t^13+18049788630144*t^14+225624841265664*t^...

>   

Guessing the series  M'(1)

>    Liste:=[seq(subs(y=1,diff(subs(x=1,co(n)),y)),n=0..80)]:

>    ser:=listtoseries(Liste,t):

>    serS2:=series(subs(t=tS,ser),S,81);

serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...
serS2 := series(4*S+12*S^2-6*S^3-66*S^4-176*S^5-454*S^6-994*S^7-1862*S^8-3438*S^9-6144*S^10-10236*S^11-16776*S^12-27484*S^13-43066*S^14-66026*S^15-103214*S^16-155878*S^17-227712*S^18-347236*S^19-512256...

>    L:=[1,serS2,series(serS2^2,S,81)]:

>    hermite_pade(L,S,80);

[-8*S-96*S^2-944*S^5+47676*S^7+16384*S^22-9308*S^12-290480*S^13-446176*S^14-392976*S^15+4096*S^24-214656*S^16-64448*S^17-13168*S^18+4096*S^21-17600*S^19-15040*S^20+223168*S^11+128672*S^8+235592*S^9+297...
[-8*S-96*S^2-944*S^5+47676*S^7+16384*S^22-9308*S^12-290480*S^13-446176*S^14-392976*S^15+4096*S^24-214656*S^16-64448*S^17-13168*S^18+4096*S^21-17600*S^19-15040*S^20+223168*S^11+128672*S^8+235592*S^9+297...
[-8*S-96*S^2-944*S^5+47676*S^7+16384*S^22-9308*S^12-290480*S^13-446176*S^14-392976*S^15+4096*S^24-214656*S^16-64448*S^17-13168*S^18+4096*S^21-17600*S^19-15040*S^20+223168*S^11+128672*S^8+235592*S^9+297...
[-8*S-96*S^2-944*S^5+47676*S^7+16384*S^22-9308*S^12-290480*S^13-446176*S^14-392976*S^15+4096*S^24-214656*S^16-64448*S^17-13168*S^18+4096*S^21-17600*S^19-15040*S^20+223168*S^11+128672*S^8+235592*S^9+297...
[-8*S-96*S^2-944*S^5+47676*S^7+16384*S^22-9308*S^12-290480*S^13-446176*S^14-392976*S^15+4096*S^24-214656*S^16-64448*S^17-13168*S^18+4096*S^21-17600*S^19-15040*S^20+223168*S^11+128672*S^8+235592*S^9+297...

The conjectured equation for M'(1)

>    algM2S:=op(1,%)+op(2,%)*M2+op(3,%)*M2^2;

algM2S := (2+182*S^3-1452*S^6+168*S^4+18*S+78*S^2-336*S^5-2156*S^7-472*S^8+4144*S^9+7752*S^10-1792*S^23+5760*S^22-5504*S^12-12864*S^13-8032*S^14+5088*S^15-3072*S^24-2048*S^25+11456*S^16+3200*S^17-7360*...
algM2S := (2+182*S^3-1452*S^6+168*S^4+18*S+78*S^2-336*S^5-2156*S^7-472*S^8+4144*S^9+7752*S^10-1792*S^23+5760*S^22-5504*S^12-12864*S^13-8032*S^14+5088*S^15-3072*S^24-2048*S^25+11456*S^16+3200*S^17-7360*...
algM2S := (2+182*S^3-1452*S^6+168*S^4+18*S+78*S^2-336*S^5-2156*S^7-472*S^8+4144*S^9+7752*S^10-1792*S^23+5760*S^22-5504*S^12-12864*S^13-8032*S^14+5088*S^15-3072*S^24-2048*S^25+11456*S^16+3200*S^17-7360*...
algM2S := (2+182*S^3-1452*S^6+168*S^4+18*S+78*S^2-336*S^5-2156*S^7-472*S^8+4144*S^9+7752*S^10-1792*S^23+5760*S^22-5504*S^12-12864*S^13-8032*S^14+5088*S^15-3072*S^24-2048*S^25+11456*S^16+3200*S^17-7360*...
algM2S := (2+182*S^3-1452*S^6+168*S^4+18*S+78*S^2-336*S^5-2156*S^7-472*S^8+4144*S^9+7752*S^10-1792*S^23+5760*S^22-5504*S^12-12864*S^13-8032*S^14+5088*S^15-3072*S^24-2048*S^25+11456*S^16+3200*S^17-7360*...

>    factor([solve(algM2S,M2)]);

[1/S^2*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1+((2*S+1)*(2*S^3-1)^2*(4*S^2+2*S+1)^3*(1+S)^6)^(1/2))*(2*S+1)/(2*S^3-1)^4, 1/S^2*(32*S^12+32*S^11+12*S^10-32...
[1/S^2*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1+((2*S+1)*(2*S^3-1)^2*(4*S^2+2*S+1)^3*(1+S)^6)^(1/2))*(2*S+1)/(2*S^3-1)^4, 1/S^2*(32*S^12+32*S^11+12*S^10-32...
[1/S^2*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1+((2*S+1)*(2*S^3-1)^2*(4*S^2+2*S+1)^3*(1+S)^6)^(1/2))*(2*S+1)/(2*S^3-1)^4, 1/S^2*(32*S^12+32*S^11+12*S^10-32...
[1/S^2*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1+((2*S+1)*(2*S^3-1)^2*(4*S^2+2*S+1)^3*(1+S)^6)^(1/2))*(2*S+1)/(2*S^3-1)^4, 1/S^2*(32*S^12+32*S^11+12*S^10-32...

The conjectured expression for M'(1)

>    M2Ssol:=(1+2*S)*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1
+(1-2*S^3)*(4*S^2+2*S+1)*(1+S)^3*sqrt(Delta))/(2*S^3-1)^4/S^2;

M2Ssol := (2*S+1)*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1+(1-2*S^3)*(4*S^2+2*S+1)*(1+S)^3*Delta^(1/2))/(2*S^3-1)^4/S^2
M2Ssol := (2*S+1)*(32*S^12+32*S^11+12*S^10-32*S^9-16*S^8+18*S^7+14*S^6-28*S^5-58*S^4-49*S^3-25*S^2-7*S-1+(1-2*S^3)*(4*S^2+2*S+1)*(1+S)^3*Delta^(1/2))/(2*S^3-1)^4/S^2

let su check the first coefficients

>    series(subs(Delta=(1+2*S)*(1+2*S+4*S^2),S=Sser,M2Ssol),t,15);

series(4*t+36*t^2+330*t^3+3178*t^4+31960*t^5+332966*t^6+3570094*t^7+39196014*t^8+438923898*t^9+4998040944*t^10+57733689964*t^11+675212545884*t^12+O(t^13),t,13)
series(4*t+36*t^2+330*t^3+3178*t^4+31960*t^5+332966*t^6+3570094*t^7+39196014*t^8+438923898*t^9+4998040944*t^10+57733689964*t^11+675212545884*t^12+O(t^13),t,13)

>    [seq(subs(y=1,diff(subs(x=1,co(n)),y)),n=0..10)];

[0, 4, 36, 330, 3178, 31960, 332966, 3570094, 39196014, 438923898, 4998040944]

>   

>   

Guessing M"(1)

>    Liste:=[seq(subs(y=1,diff(subs(x=1,co(n)),y$2)),n=0..80)]:

>    ser:=listtoseries(Liste,t):

>    serS3:=series(subs(t=tS,ser),S,81);

serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...
serS3 := series(4*S+76*S^2+320*S^3+356*S^4-1244*S^5-7464*S^6-25616*S^7-71832*S^8-175292*S^9-389216*S^10-813640*S^11-1617992*S^12-3077500*S^13-5668624*S^14-10169984*S^15-17761960*S^16-30384588*S^17-5115...

>    L:=[1,serS2,serS3]:

>    factor(hermite_pade(L,S,81));

[-4*S*(2*S+1)*(32*S^16+160*S^15+336*S^14-160*S^13-2072*S^12-4752*S^11-5952*S^10-3824*S^9+428*S^8+3804*S^7+4388*S^6+2890*S^5+1138*S^4+174*S^3-70*S^2-47*S-10), 2*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-...
[-4*S*(2*S+1)*(32*S^16+160*S^15+336*S^14-160*S^13-2072*S^12-4752*S^11-5952*S^10-3824*S^9+428*S^8+3804*S^7+4388*S^6+2890*S^5+1138*S^4+174*S^3-70*S^2-47*S-10), 2*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-...

Conjectured equation for M"(1)

>    algM2M3S:=op(1,%)+op(2,%)*M2+op(3,%)*M3;

algM2M3S := -4*S*(2*S+1)*(32*S^16+160*S^15+336*S^14-160*S^13-2072*S^12-4752*S^11-5952*S^10-3824*S^9+428*S^8+3804*S^7+4388*S^6+2890*S^5+1138*S^4+174*S^3-70*S^2-47*S-10)+2*(8*S^7+8*S^6+12*S^5-20*S^4-48*S...
algM2M3S := -4*S*(2*S+1)*(32*S^16+160*S^15+336*S^14-160*S^13-2072*S^12-4752*S^11-5952*S^10-3824*S^9+428*S^8+3804*S^7+4388*S^6+2890*S^5+1138*S^4+174*S^3-70*S^2-47*S-10)+2*(8*S^7+8*S^6+12*S^5-20*S^4-48*S...

Conjectured expression for M"(1)

>    M3Ssol:=collect(solve(subs(M2=M2Ssol,algM2M3S),M3),Delta,factor);

M3Ssol := 2*(2*S+1)*(4*S^2+2*S+1)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(1+S)^3/(2*S^3-1)^5/S^3*Delta^(1/2)-2*(2*S+1)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S...
M3Ssol := 2*(2*S+1)*(4*S^2+2*S+1)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(1+S)^3/(2*S^3-1)^5/S^3*Delta^(1/2)-2*(2*S+1)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S...
M3Ssol := 2*(2*S+1)*(4*S^2+2*S+1)*(8*S^7+8*S^6+12*S^5-20*S^4-48*S^3-42*S^2-19*S-5)*(1+S)^3/(2*S^3-1)^5/S^3*Delta^(1/2)-2*(2*S+1)*(192*S^19+192*S^18+64*S^17-96*S^16+1728*S^15+6016*S^14+10080*S^13+8336*S...

Let us check

>    series(subs(Delta=(1+2*S)*(1+2*S+4*S^2),S=Sser,M3Ssol),t,15);

series(4*t+100*t^2+1424*t^3+17916*t^4+216612*t^5+2585808*t^6+30804752*t^7+367913056*t^8+4414218004*t^9+53247259096*t^10+645914804440*t^11+O(t^12),t,12)

>    [seq(subs(y=1,diff(subs(x=1,co(n)),y$2)),n=0..10)];

[0, 4, 100, 1424, 17916, 216612, 2585808, 30804752, 367913056, 4414218004, 53247259096]

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