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A370153 Expansion of g.f. (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)).

%I A370153 #23 Feb 16 2025 08:34:06
%S A370153 1,0,-1,1,1,-1,-2,1,4,-2,-5,3,6,-4,-9,6,13,-8,-17,11,21,-15,-28,19,39,
%T A370153 -25,-49,33,60,-42,-78,53,101,-68,-125,87,153,-108,-192,134,241,-167,
%U A370153 -295,207,357,-255,-438,311,540,-380,-652,465,781,-563,-946,678,1145,-819,-1368,986,1627
%N A370153 Expansion of g.f. (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2)) / (1 - x^(4*n)).
%C A370153 Column 0 of triangle A370041. The g.f. of triangle A370041, G(x,y), satisfies Sum_{n=-oo..+oo} (x^n - y*G(x,y))^n = 1 - (y-2)*Sum_{n>=1} x^(n^2). The g.f. of this sequence is G(x,y) at y = 0.
%H A370153 Paul D. Hanna, <a href="/A370153/b370153.txt">Table of n, a(n) for n = 1..201</a>
%H A370153 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370153 a(n) = A370041(n,0) for n >= 1.
%F A370153 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370153 (1) A(x) = (theta_3(x) - 1)/2 * Product_{n>=1} (1 - x^(4*n-2))/(1 - x^(4*n)).
%F A370153 (2) A(x) = P(x)/Q(x) where P(x) = Sum_{n>=1} x^(n^2) and Q(x) = Sum_{n>=0} x^(n*(n+1)).
%F A370153 (3) A(x) = G(x,0) where G(x,y) is the g.f. of triangle A370041 (see comment).
%e A370153 G.f.: A(x) = x - x^3 + x^4 + x^5 - x^6 - 2*x^7 + x^8 + 4*x^9 - 2*x^10 - 5*x^11 + 3*x^12 + 6*x^13 - 4*x^14 - 9*x^15 + 6*x^16 + 13*x^17 - 8*x^18 + ...
%e A370153 which equals A(x) = P(x) / Q(x)
%e A370153 where
%e A370153 P(x) = x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
%e A370153 Q(x) = 1 + x^2 + x^6 + x^12 + x^20 + x^30 + x^42 + ...
%o A370153 (PARI) {a(n) = my(P = sum(m=1,sqrtint(n+1), x^(m^2) +x*O(x^n)),
%o A370153 Q = sum(m=0,sqrtint(n+1), x^(m*(m+1)) +x*O(x^n))); polcoeff(P/Q,n)}
%o A370153 for(n=1,50,print1(a(n),", "))
%Y A370153 Cf. A370041, A370154, A370155.
%Y A370153 Cf. A370150 (dual).
%K A370153 sign
%O A370153 1,7
%A A370153 _Paul D. Hanna_, Feb 10 2024