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 A370021 #14 Feb 16 2025 08:34:06
%S A370021 1,1,4,9,22,63,155,415,1124,2957,8047,21817,59048,161870,442675,
%T A370021 1214563,3348145,9228858,25514319,70682731,195993889,544578231,
%U A370021 1515027660,4219560585,11768353857,32853953466,91812137378,256831830373,719046731299,2014808160498,5650037329385
%N A370021 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370021 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370021 a(n+1)/a(n) tends to 2.874307... - _Vaclav Kotesovec_, Feb 11 2024
%H A370021 Paul D. Hanna, <a href="/A370021/b370021.txt">Table of n, a(n) for n = 1..401</a>
%H A370021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370021 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370021 (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370021 (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^(n-1) = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370021 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^n = 0.
%F A370021 (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^n)^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370021 (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^n)^(n+1) = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370021 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + A(x)*x^n)^(n+1) = 0.
%e A370021 G.f.: A(x) = x + x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 63*x^6 + 155*x^7 + 415*x^8 + 1124*x^9 + 2957*x^10 + 8047*x^11 + 21817*x^12 + ...
%e A370021 where
%e A370021 Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 - 3*x + 3*x^4 - 3*x^9 + 3*x^16 - 3*x^25 + 3*x^36 - 3*x^49 +- ...
%e A370021 SPECIAL VALUES.
%e A370021 (V.1) Let A = A(exp(-Pi)) = 0.04543932020196352081239499480519595850147996376296857684...
%e A370021 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + A)^n = (3*(Pi/2)^(1/4)/gamma(3/4) - 1)/2 = 0.87036870723417523211086...
%e A370021 (V.2) Let A = A(exp(-2*Pi)) = 0.00187095623366907901234297087932572258706353074482100743...
%e A370021 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + A)^n = (3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 1)/2 = 0.99626511456090713578995...
%e A370021 (V.3) Let A = A(-exp(-Pi)) = -0.04164083178192506029717066967023726841141127226704810579...
%e A370021 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + A)^n = (3*Pi^(1/4)/gamma(3/4) - 1)/2 = 1.12965221681996202186297...
%e A370021 (V.4) Let A = A(-exp(-2*Pi)) = -0.00186398133004329627873834535037664668964585574963215266...
%e A370021 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + A)^n = (3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 1)/2 = 1.00560232823160863657151...
%o A370021 (PARI) {a(n) = my(A=[0,1]); for(i=0,n, A = concat(A,0);
%o A370021 A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 1*Ser(A))^m ) - 1 - 3*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1) ); A[n+1]}
%o A370021 for(n=1,30, print1(a(n),", "))
%Y A370021 Cf. A370020, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029.
%K A370021 nonn
%O A370021 1,3
%A A370021 _Paul D. Hanna_, Feb 09 2024