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A359916 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).

%I A359916 #15 Jan 05 2024 16:36:25
%S A359916 1,1,7,48,349,2718,22403,192375,1701544,15389227,141643233,1322344998,
%T A359916 12491424723,119177917679,1146750961711,11115577075944,
%U A359916 108437559699613,1063849149587086,10489551647580027,103891138998923739,1033113794091793406,10310925888014393461
%N A359916 a(n) = coefficient of x^n in A(x) such that A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).
%H A359916 Paul D. Hanna, <a href="/A359916/b359916.txt">Table of n, a(n) for n = 1..200</a>
%H A359916 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuintupleProductIdentity.html">Quintuple Product Identity</a>.
%F A359916 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following.
%F A359916 (1) A(x) = 1 + Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (1/A(x)^(3*n) - A(x)^(3*n+1)).
%F A359916 (2) -1 = Product_{n>=1} (1 - x^n) * (1 - x^n*A(x)) * (1 - x^n/A(x)) * (1 - x^(2*n-1)*A(x)^2) * (1 - x^(2*n-1)/A(x)^2), by the Watson quintuple product identity.
%e A359916 G.f.: A(x) = x + x^2 + 7*x^3 + 48*x^4 + 349*x^5 + 2718*x^6 + 22403*x^7 + 192375*x^8 + 1701544*x^9 + 15389227*x^10 + ...
%e A359916 where A = A(x) satisfies the doubly infinite series
%e A359916 A(x) - 1 = ... + x^12*(A^9 - 1/A^8) + x^5*(A^6 - 1/A^5) + x*(A^3 - 1/A^2) + (1 - A) + x^2*(1/A^3 - A^4) + x^7*(1/A^6 - A^7) + x^15*(1/A^9 - A^10) + ... + x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)) + ...
%e A359916 also, by the Watson quintuple product identity,
%e A359916 -1 = (1-x^1)*(1-x^1*A)*(1-x^1/A)*(1-x^1*A^2)*(1-x^1/A^2) * (1-x^2)*(1-x^2*A)*(1-x^2/A)*(1-x^3*A^2)*(1-x^3/A^2) * (1-x^3)*(1-x^3*A)*(1-x^3/A)*(1-x^5*A^2)*(1-x^5/A^2) * (1-x^4)*(1-x^4*A)*(1-x^4/A)*(1-x^7*A^2)*(1-x^7/A^2) * ...
%o A359916 (PARI) /* Using the doubly infinite series */
%o A359916 {a(n) = my(A=[0,1,1]); for(i=1,n, A = concat(A,0);
%o A359916 A[#A] = polcoeff(1 - Ser(A) + sum(m=-#A, #A, x^(m*(3*m+1)/2) * (1/Ser(A)^(3*m) - Ser(A)^(3*m+1)) ),#A-3) ); A[n+1]}
%o A359916 for(n=1,30, print1(a(n),", "))
%o A359916 (PARI) /* Using the quintuple product */
%o A359916 {a(n) = my(A=[0,1,1]); for(i=1,n, A = concat(A,0);
%o A359916 A[#A] = polcoeff(1 + prod(m=1,#A, (1 - x^m) * (1 - x^m*Ser(A)) * (1 - x^m/Ser(A)) * (1 - x^(2*m-1)*Ser(A)^2) * (1 - x^(2*m-1)/Ser(A)^2)),#A-3) ); A[n+1]}
%o A359916 for(n=1,30, print1(a(n),", "))
%Y A359916 Cf. A359915, A359920, A359919, A359921, A359924.
%Y A359916 Cf. A359914.
%K A359916 nonn
%O A359916 1,3
%A A359916 _Paul D. Hanna_, Jan 22 2023