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 A370024 #10 Feb 16 2025 08:34:06
%S A370024 1,4,19,99,529,2853,15566,85879,477716,2674070,15047671,85063429,
%T A370024 482733230,2748703604,15697194139,89875431754,515774659357,
%U A370024 2966016776556,17088046518051,98614323921685,569967829487533,3298876334401503,19117753534875276,110922240116613681,644276475406441599
%N A370024 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370024 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%H A370024 Paul D. Hanna, <a href="/A370024/b370024.txt">Table of n, a(n) for n = 1..401</a>
%H A370024 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370024 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370024 (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370024 (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^(n-1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370024 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^n = 0.
%F A370024 (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370024 (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^(n+1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370024 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 4*A(x)*x^n)^(n+1) = 0.
%e A370024 G.f.: A(x) = x + 4*x^2 + 19*x^3 + 99*x^4 + 529*x^5 + 2853*x^6 + 15566*x^7 + 85879*x^8 + 477716*x^9 + 2674070*x^10 + 15047671*x^11 + 85063429*x^12 + ...
%e A370024 where
%e A370024 Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 - 6*x + 6*x^4 - 6*x^9 + 6*x^16 - 6*x^25 + 6*x^36 - 6*x^49 +- ...
%e A370024 SPECIAL VALUES.
%e A370024 (V.1) Let A = A(exp(-Pi)) = 0.05266628449954901094912490050067062239110765179054552678...
%e A370024 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 4*A)^n = 3*(Pi/2)^(1/4)/gamma(3/4) - 2 = 0.74073741446835046422172...
%e A370024 (V.2) Let A = A(exp(-2*Pi)) = 0.001881517053093894919707587041659521876650213322334450878...
%e A370024 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 4*A)^n = 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 2 = 0.988795343682721407369...
%e A370024 (V.3) Let A = A(-exp(-Pi)) = -0.03699687105031477666227946508842289849689211763245984347...
%e A370024 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 4*A)^n = 3*Pi^(1/4)/gamma(3/4) - 2 = 1.259304433639924043725...
%e A370024 (V.4) Let A = A(-exp(-2*Pi)) = -0.0018536159060139689998658447922411419770684746756327600438...
%e A370024 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 4*A)^n = 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 2 = 1.011204656463217273143...
%o A370024 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370024 A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 4*Ser(A))^m ) - 1 - 6*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/4 ); A[n+1]}
%o A370024 for(n=1,30, print1(a(n),", "))
%Y A370024 Cf. A370020, A370021, A370022, A370023, A370025, A370026, A370027, A370028, A370029, A370042.
%K A370024 nonn
%O A370024 1,2
%A A370024 _Paul D. Hanna_, Feb 09 2024