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 A370027 #11 Feb 16 2025 08:34:06
%S A370027 1,7,52,405,3250,26541,219311,1828657,15360068,129802889,1102476535,
%T A370027 9403920685,80507808128,691425600548,5954703569335,51409228587355,
%U A370027 444806083780093,3856115167020090,33488422645226379,291294693699275917,2537471770952346625,22133307405655321131
%N A370027 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
%C A370027 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%H A370027 Paul D. Hanna, <a href="/A370027/b370027.txt">Table of n, a(n) for n = 1..326</a>
%H A370027 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370027 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370027 (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370027 (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 7*A(x))^(n-1) = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370027 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 7*A(x))^n = 0.
%F A370027 (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 7*A(x)*x^n)^n = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370027 (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 7*A(x)*x^n)^(n+1) = 1 + 9*Sum_{n>=1} (-1)^n * x^(n^2).
%F A370027 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 7*A(x)*x^n)^(n+1) = 0.
%e A370027 G.f.: A(x) = x + 7*x^2 + 52*x^3 + 405*x^4 + 3250*x^5 + 26541*x^6 + 219311*x^7 + 1828657*x^8 + 15360068*x^9 + 129802889*x^10 + 1102476535*x^11 + 9403920685*x^12 + ...
%e A370027 where
%e A370027 Sum_{n=-oo..+oo} (-1)^n * (x^n + 7*A(x))^n = 1 - 9*x + 9*x^4 - 9*x^9 + 9*x^16 - 9*x^25 + 9*x^36 - 9*x^49 +- ...
%e A370027 SPECIAL VALUES.
%e A370027 (V.1) Let A = A(exp(-Pi)) = 0.06265408791983395104830182276472061307372169283289177444...
%e A370027 then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 7*A)^n = (9*(Pi/2)^(1/4)/gamma(3/4) - 7)/2 = 0.6111061217025256963...
%e A370027 (V.2) Let A = A(exp(-2*Pi)) = 0.001892197774017068345453024031418945825808997896316975979...
%e A370027 then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 7*A)^n = (9*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 7)/2 = 0.98319301552408211105...
%e A370027 (V.3) Let A = A(-exp(-Pi)) = -0.03328815108533045197898037729675109506494860109014140530...
%e A370027 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 7*A)^n = (9*Pi^(1/4)/gamma(3/4) - 7)/2 = 1.388956650459886065588...
%e A370027 (V.4) Let A = A(-exp(-2*Pi)) = -0.001843365127917378852723125074532830028319143070315792225...
%e A370027 then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 7*A)^n = (9*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 7)/2 = 1.0168069846948259097...
%o A370027 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370027 A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 7*Ser(A))^m ) - 1 - 9*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/7 ); A[n+1]}
%o A370027 for(n=1,30, print1(a(n),", "))
%Y A370027 Cf. A370020, A370021, A370022, A370023, A370024, A370025, A370026, A370028, A370029.
%K A370027 nonn
%O A370027 1,2
%A A370027 _Paul D. Hanna_, Feb 09 2024