cp's OEIS Frontend

A370150 Expansion of g.f. (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.

%I A370150 #22 Feb 16 2025 08:34:06
%S A370150 1,0,3,-1,9,-3,22,-9,52,-22,111,-51,230,-108,451,-222,861,-432,1587,
%T A370150 -819,2861,-1501,5028,-2691,8679,-4707,14691,-8089,24492,-13638,40202,
%U A370150 -22653,65141,-37060,104199,-59863,164833,-95484,257920,-150646,399681,-235141,613557,-363543,933869
%N A370150 Expansion of g.f. (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.
%C A370150 Column 0 of triangle A370040. The g.f. of triangle A370040, F(x,y), satisfies Sum_{n=-oo..+oo} (-1)^n * (x^n + y*F(x,y))^n = 1 + (y+2)*Sum_{n>=1} (-1)^n * x^(n^2). The g.f. of this sequence is F(x,y) at y = 0.
%H A370150 Paul D. Hanna, <a href="/A370150/b370150.txt">Table of n, a(n) for n = 1..202</a>
%H A370150 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370150 a(n) = A370040(n,0) for n >= 1.
%F A370150 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370150 (1) A(x) = (1 - theta_4(x))/2 / Product_{n>=1} (1 - x^(2*n))^3.
%F A370150 (2) A(x) = P(x)/Q(x) where P(x) = Sum_{n>=1} (-1)^(n-1) * x^(n^2) and Q(x) = Sum_{n>=0} (-1)^n * (2*n+1) * x^(n*(n+1)).
%F A370150 (3) A(x) = F(x,0) where F(x,y) is the g.f. of triangle A370040 (see comment).
%e A370150 G.f.: A(x) = x + 3*x^3 - x^4 + 9*x^5 - 3*x^6 + 22*x^7 - 9*x^8 + 52*x^9 - 22*x^10 + 111*x^11 - 51*x^12 + 230*x^13 - 108*x^14 + 451*x^15 - 222*x^16 + ...
%e A370150 which equals A(x) = P(x) / Q(x)
%e A370150 where
%e A370150 P(x) = x - x^4 + x^9 - x^16 + x^25 - x^36 + x^49 + ...
%e A370150 Q(x) = 1 - 3*x^2 + 5*x^6 - 7*x^12 + 9*x^20 - 11*x^30 + 13*x^42 + ...
%o A370150 (PARI) {a(n) = my(P = sum(m=1,sqrtint(n+1), (-1)^(m-1) * x^(m^2) +x*O(x^n)),
%o A370150 Q = sum(m=0,sqrtint(n+1), (-1)^m * (2*m+1) * x^(m*(m+1)) +x*O(x^n)));
%o A370150 polcoeff(P/Q,n)}
%o A370150 for(n=1,50,print1(a(n),", "))
%Y A370150 Cf. A370040, A370151, A370152.
%Y A370150 Cf. A370153 (dual).
%K A370150 sign
%O A370150 1,3
%A A370150 _Paul D. Hanna_, Feb 10 2024