%I #14 Jan 18 2024 14:01:18

%S 1,3,24,171,1335,11940,115773,1160901,11901537,124726644,1332688035,

%T 14455451526,158660036535,1758835084221,19667067522966,

%U 221573079684087,2512635069594897,28656903391830291,328500210705228867,3782806859877522522,43738575934977450465

%N Expansion of g.f. A(x) satisfying 3*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

%H Paul D. Hanna, <a href="/A361553/b361553.txt">Table of n, a(n) for n = 0..300</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.

%F G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.

%F (1) 3*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).

%F (2) 3*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).

%F (3) 3*x = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^(n-1)/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.

%F (4) a(n) = Sum_{k=0..n} A361550(n,k) * 3^k for n >= 0.

%F a(n) ~ c * d^n / n^(3/2), where d = 12.47776743014414138089586... and c = 0.474320402676760199022... - _Vaclav Kotesovec_, Mar 29 2023

%e G.f.: A(x) = 1 + 3*x + 24*x^2 + 171*x^3 + 1335*x^4 + 11940*x^5 + 115773*x^6 + 1160901*x^7 + 11901537*x^8 + 124726644*x^9 + ...

%e where A = A(x) satisfies the doubly infinite sum

%e 3*x = ... + x^12*(1/A^9 - A^8) + x^5*(1/A^6 - A^5) + x*(1/A^3 - A^2) + (1 - 1/A) + x^2*(A^3 - 1/A^4) + x^7*(A^6 - 1/A^7) + x^15*(A^9 - 1/A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...

%e also, by the Watson quintuple product identity,

%e 3*x = (1-x)*(1-x*A)*(1-1/A)*(1-x*A^2)*(1-x/A^2) * (1-x^2)*(1-x^2*A)*(1-x/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^2/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^3/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...

%t (* Calculation of constant d: *) With[{k = 3}, 1/r /. FindRoot[{r^3*s^3 * QPochhammer[r] * QPochhammer[1/(r*s^2), r^2] * QPochhammer[1/(r*s), r] * QPochhammer[s, r] * QPochhammer[s^2/r, r^2] / ((-1 + s)*(-1 + r*s)*(-r + s^2)*(-1 + r*s^2)) == k*r, 1/(-1 + s) + 1/(s*(-1 + r*s)) + (2*s)/(-r + s^2) - 2/(s - r*s^3) + (-QPolyGamma[0, -Log[r*s]/Log[r], r] + QPolyGamma[0, Log[s]/Log[r], r] - QPolyGamma[0, -Log[r*s^2]/Log[r^2], r^2] + QPolyGamma[0, Log[s^2/r]/Log[r^2], r^2]) / (s*Log[r]) == 0}, {r, 1/12}, {s, 2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Jan 18 2024 *)

%o (PARI) /* Using the doubly infinite series */

%o {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff(3*x - sum(m=-#A, #A, x^(m*(3*m-1)/2) * Ser(A)^(3*m-1) * (x^m*Ser(A) - 1) ) , #A-1) ); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%o (PARI) /* Using the quintuple product */

%o {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff(3*x - prod(m=1, #A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^(m-1)/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2) ), #A-1) ); A[n+1]}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A361550, A359920, A361552, A361554, A361555.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 19 2023