This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A361552 #14 Jan 18 2024 14:00:07
%S A361552 1,2,14,84,530,3770,29446,240302,2003914,17024332,147306448,
%T A361552 1294859540,11524690228,103605031978,939357512086,8580744729478,
%U A361552 78898896072996,729661925134886,6782435427053490,63332055630823770,593793935288453260,5587934788557993846
%N A361552 Expansion of g.f. A(x) satisfying 2*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
%C A361552 Conjecture: a(n) == A359914(n-1) (mod 4) for n > 1.
%C A361552 Conjecture: a(n)/2 == A000041(n-1) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n-1, for n >= 1.
%H A361552 Paul D. Hanna, <a href="/A361552/b361552.txt">Table of n, a(n) for n = 0..300</a>
%H A361552 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.
%F A361552 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A361552 (1) 2*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
%F A361552 (2) 2*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
%F A361552 (3) 2*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 A361552 (4) a(n) = Sum_{k=0..n} A361550(n,k) * 2^k for n >= 0.
%F A361552 a(n) ~ c * d^n / n^(3/2), where d = 10.118828419885936478438... and c = 0.4250308979334609908... - _Vaclav Kotesovec_, Mar 19 2023
%e A361552 G.f.: A(x) = 1 + 2*x + 14*x^2 + 84*x^3 + 530*x^4 + 3770*x^5 + 29446*x^6 + 240302*x^7 + 2003914*x^8 + 17024332*x^9 + ...
%e A361552 where A = A(x) satisfies the doubly infinite sum
%e A361552 2*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 A361552 also, by the Watson quintuple product identity,
%e A361552 2*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 A361552 (* Calculation of constant d: *) With[{k = 2}, 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, 1/10}, {s, 2}, WorkingPrecision -> 70]] (* _Vaclav Kotesovec_, Jan 18 2024 *)
%o A361552 (PARI) /* Using the doubly infinite series */
%o A361552 {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A361552 A[#A] = polcoeff(2*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 A361552 for(n=0, 30, print1(a(n), ", "))
%o A361552 (PARI) /* Using the quintuple product */
%o A361552 {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0);
%o A361552 A[#A] = polcoeff(2*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 A361552 for(n=0, 30, print1(a(n), ", "))
%Y A361552 Cf. A361550, A359920, A361553, A361554, A361555.
%Y A361552 Cf. A359914, A040051, A000041.
%K A361552 nonn
%O A361552 0,2
%A A361552 _Paul D. Hanna_, Mar 19 2023