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 A370023 #10 Feb 16 2025 08:34:06
%S A370023 1,3,12,53,234,1041,4711,21573,99484,461657,2154591,10102701,47555840,
%T A370023 224624016,1064183887,5055060411,24068888061,114841741098,
%U A370023 548992775523,2628924592737,12608597616161,60558351876803,291238387762452,1402314223189959,6759651098793285,32617445956236720
%N A370023 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370023 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%H A370023 Paul D. Hanna, <a href="/A370023/b370023.txt">Table of n, a(n) for n = 1..401</a>
%H A370023 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370023 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370023 (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370023 (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^(n-1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370023 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^n = 0.
%F A370023 (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370023 (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^(n+1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370023 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 3*A(x)*x^n)^(n+1) = 0.
%e A370023 G.f.: A(x) = x + 3*x^2 + 12*x^3 + 53*x^4 + 234*x^5 + 1041*x^6 + 4711*x^7 + 21573*x^8 + 99484*x^9 + 461657*x^10 + 2154591*x^11 + 10102701*x^12 + ...
%e A370023 where
%e A370023 Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 - 5*x + 5*x^4 - 5*x^9 + 5*x^16 - 5*x^25 + 5*x^36 - 5*x^49 +- ...
%e A370023 SPECIAL VALUES.
%e A370023 (V.1) Let A = A(exp(-Pi)) = 0.05001316702398359971645418498866690386932728399152644693...
%e A370023 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 3*A)^n = (5*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.78394784539029205351810...
%e A370023 (V.2) Let A = A(exp(-2*Pi)) = 0.001877983557643657576778844718492775838546798118866577860...
%e A370023 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 3*A)^n = (5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.990662786402267839474...
%e A370023 (V.3) Let A = A(-exp(-Pi)) = -0.03842474691590612761867206263978602696713545771404819339...
%e A370023 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 3*A)^n = (5*Pi^(1/4)/gamma(3/4) - 3)/2 = 1.216087028033270036438...
%e A370023 (V.4) Let A = A(-exp(-2*Pi)) = -0.001857058214293085256892081751882664927312970576990961749...
%e A370023 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 3*A)^n = (5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3)/2 = 1.009337213719347727619...
%o A370023 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370023 A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 3*Ser(A))^m ) - 1 - 5*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/3 ); A[n+1]}
%o A370023 for(n=1,30, print1(a(n),", "))
%Y A370023 Cf. A370020, A370021, A370022, A370024, A370025, A370026, A370027, A370028, A370029, A370042.
%K A370023 nonn
%O A370023 1,2
%A A370023 _Paul D. Hanna_, Feb 09 2024