This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370022 #14 Feb 16 2025 08:34:06
%S A370022 1,2,7,25,85,301,1086,3927,14328,52724,194915,723845,2699878,10104968,
%T A370022 37933855,142795810,538829973,2037596590,7720231359,29302685197,
%U A370022 111398230285,424115408181,1616860117052,6171586558551,23583939930835,90218328876825,345461395176495,1324041033133129
%N A370022 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370022 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%H A370022 Paul D. Hanna, <a href="/A370022/b370022.txt">Table of n, a(n) for n = 1..401</a>
%H A370022 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370022 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370022 (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370022 (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 2*A(x))^(n-1) = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370022 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 2*A(x))^n = 0.
%F A370022 (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^n)^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370022 (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^n)^(n+1) = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370022 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 2*A(x)*x^n)^(n+1) = 0.
%e A370022 G.f.: A(x) = x + 2*x^2 + 7*x^3 + 25*x^4 + 85*x^5 + 301*x^6 + 1086*x^7 + 3927*x^8 + 14328*x^9 + 52724*x^10 + 194915*x^11 + 723845*x^12 + ...
%e A370022 where
%e A370022 Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 - 4*x + 4*x^4 - 4*x^9 + 4*x^16 - 4*x^25 + 4*x^36 - 4*x^49 +- ...
%e A370022 SPECIAL VALUES.
%e A370022 (V.1) Let A = A(exp(-Pi)) = 0.04761601613534030259384050896565071457116692089742172541...
%e A370022 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 2*A)^n = 2*(Pi/2)^(1/4)/gamma(3/4) - 1 = 0.82715827631223364281448...
%e A370022 (V.2) Let A = A(exp(-2*Pi)) = 0.00187446330928756547025110339586987296984387228299321603...
%e A370022 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 2*A)^n = 2*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 1 = 0.99253022912181427157991...
%e A370022 (V.3) Let A = A(-exp(-Pi)) = -0.03996785964385216049635981950386915887875531406265280233...
%e A370022 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 2*A)^n = 2*Pi^(1/4)/gamma(3/4) - 1 = 1.1728696224266160291506...
%e A370022 (V.4) Let A = A(-exp(-2*Pi)) = -0.00186051333175936112600864666861119312780357024086759004...
%e A370022 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 2*A)^n = 2*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 1 = 1.007469770975478182...
%o A370022 (PARI) {a(n) = my(A=[0,1]); for(i=0,n, A = concat(A,0);
%o A370022 A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 2*Ser(A))^m ) - 1 - 4*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/2 ); A[n+1]}
%o A370022 for(n=1,30, print1(a(n),", "))
%Y A370022 Cf. A370020, A370021, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.
%K A370022 nonn
%O A370022 1,2
%A A370022 _Paul D. Hanna_, Feb 09 2024