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A354651 G.f. A(x) satisfies: 1/(1 - x) = Sum_{n>=1} (-1)^(n-1) * A(x)^(n^2).

%I A354651 #17 Feb 16 2025 08:34:03
%S A354651 1,1,1,2,5,11,25,64,168,434,1136,3046,8246,22400,61290,169036,468628,
%T A354651 1304390,3646104,10232796,28814306,81376616,230462906,654363034,
%U A354651 1862260359,5311064061,15176758091,43448083792,124593820615,357853635931,1029326055479,2964817204082
%N A354651 G.f. A(x) satisfies: 1/(1 - x) = Sum_{n>=1} (-1)^(n-1) * A(x)^(n^2).
%C A354651 Conjectures:
%C A354651 (C.1) a(4*n) = 0 (mod 2) for n >= 0.
%C A354651 (C.2) a(4*n+1) = a(4*n+2) = a(4*n+3) (mod 2) for n >= 0.
%C A354651 (C.3) a(4*n+1) = a(4*n+3) (mod 4) for n >= 0.
%H A354651 Paul D. Hanna, <a href="/A354651/b354651.txt">Table of n, a(n) for n = 1..625</a>
%H A354651 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A354651 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies:
%F A354651 (1) (1 - 3*x)/(1 - x) = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n^2).
%F A354651 (2) (1 - 3*x)/(1 - x) = Product_{n>=1} (1 - A(x)^(2*n)) * (1 - A(x)^(2*n-1))^2, by the Jacobi triple product identity.
%F A354651 (3) (1 - 3*x)^2/(1 - x)^2 = 1 + 4*Sum_{n>=1} (-1)^n * A(x)^(2*n-1)/(1 + A(x)^(2*n-1)), by a q-series identity for the Jacobi theta_3 function.
%F A354651 (4) (1 - 3*x)^4/(1 - x)^4 = 1 + 8*Sum_{n>=1} (-1)^n * n * A(x)^n/(1 + A(x)^n), by a q-series identity for the Jacobi theta_3 function.
%e A354651 G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 25*x^7 + 64*x^8 + 168*x^9 + 434*x^10 + 1136*x^11 + 3046*x^12 + 8246*x^13 + 22400*x^14 + ...
%e A354651 where
%e A354651 1/(1-x) = A(x) - A(x)^4 + A(x)^9 - A(x)^16 + A(x)^25 - A(x)^36 + A(x)^49 -+ ... + (-1)^(n-1) * A(x)^(n^2) + ...
%e A354651 By the Jacobi triple product
%e A354651 (1 - 3*x)/(1 - x) = (1 - A(x)^2)*(1 - A(x))^2 * (1 - A(x)^4) * (1 - A(x)^3)^2 * (1 - A(x)^6) * (1 - A(x)^5)^2 * (1 - A(x)^8) * (1 - A(x)^7)^2 * ...
%o A354651 (PARI) {a(n) = my(A=[0,1],t); for(i=1,n, A = concat(A,0); t = sqrtint(#A)+1;
%o A354651 A[#A] = 1 + polcoeff( sum(n=1,t, (-1)^n * Ser(A)^(n^2)), #A-1)); H=A; A[n+1]}
%o A354651 for(n=1,40,print1(a(n),", "))
%Y A354651 Cf. A006456 (the series reversion of -A(-x) is the g.f. for A006456, apart from the initial term).
%Y A354651 Cf. A355151.
%K A354651 nonn
%O A354651 1,4
%A A354651 _Paul D. Hanna_, Jun 18 2022