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 A370042 #11 Feb 16 2025 08:34:06
%S A370042 1,10,103,1089,11749,128637,1423982,15898231,178717112,2020360748,
%T A370042 22947819571,261696375829,2994717484790,34373295184712,
%U A370042 395580223408591,4563146810297938,52747346257279381,610871638149166758,7086520419499114527,82334442066436896541,957935578573905521101
%N A370042 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370042 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%H A370042 Paul D. Hanna, <a href="/A370042/b370042.txt">Table of n, a(n) for n = 1..272</a>
%H A370042 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370042 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370042 (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370042 (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^(n-1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370042 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 10*A(x))^n = 0.
%F A370042 (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^n = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370042 (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 10*A(x)*x^n)^(n+1) = 1 + 12*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370042 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 10*A(x)*x^n)^(n+1) = 0.
%e A370042 G.f.: A(x) = x + 10*x^2 + 103*x^3 + 1089*x^4 + 11749*x^5 + 128637*x^6 + 1423982*x^7 + 15898231*x^8 + 178717112*x^9 + 2020360748*x^10 + 22947819571*x^11 + ...
%e A370042 where
%e A370042 Sum_{n=-oo..+oo} (-1)^n * (x^n + 10*A(x))^n = 1 - 12*x + 12*x^4 - 12*x^9 + 12*x^16 - 12*x^25 + 12*x^36 - 12*x^49 +- ...
%e A370042 SPECIAL VALUES.
%e A370042 (V.1) Let A = A(exp(-Pi)) = 0.07738488286995169642543180751945321776018365032150702566...
%e A370042 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 10*A)^n = 6*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.4814748289367009284...
%e A370042 (V.2) Let A = A(exp(-2*Pi)) = 0.001903000450057888437867399675031908155434474357834107336...
%e A370042 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 10*A)^n = 6*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5 = 0.9775906873654428147...
%e A370042 (V.3) Let A = A(-exp(-Pi)) = -0.03025721520362353256298796517975081121112509387406260314...
%e A370042 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 10*A)^n = 6*Pi^(1/4)/gamma(3/4) - 5 = 1.518608867279848087...
%e A370042 (V.4) Let A = A(-exp(-2*Pi)) = -0.001833227104147111248063467420834469150941590720555554025...
%e A370042 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 10*A)^n = 6*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5 = 1.0224093129264345...
%o A370042 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370042 A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 10*Ser(A))^m ) - 1 - 12*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/10 ); A[n+1]}
%o A370042 for(n=1,30, print1(a(n),", "))
%Y A370042 Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029.
%K A370042 nonn
%O A370042 1,2
%A A370042 _Paul D. Hanna_, Feb 08 2024