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%I #12 Jan 05 2024 23:32:55
%S 1,4,36,296,2732,28980,329996,3872908,46575260,573472248,7197096168,
%T 91640952360,1180636398320,15364364313588,201691201775092,
%U 2667523242203932,35510152549696208,475424653523498396,6397601663340197268,86481499341290372804,1173813146742741571560
%N Expansion of g.f. A(x) satisfying 4*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="/A361554/b361554.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) 4*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).
%F (2) 4*x = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) * A(x)^(3*n) * (x^n - 1/A(x)).
%F (3) 4*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) * 4^k for n >= 0.
%e G.f.: A(x) = 1 + 4*x + 36*x^2 + 296*x^3 + 2732*x^4 + 28980*x^5 + 329996*x^6 + 3872908*x^7 + 46575260*x^8 + 573472248*x^9 + ...
%e where A = A(x) satisfies the doubly infinite sum
%e 4*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 4*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) * ...
%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(4*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(4*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, A361553, A361555.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Mar 19 2023
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