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 A369671 #21 Feb 16 2025 08:34:06
%S A369671 1,4,15,52,177,664,3038,16268,90660,490456,2541387,12819184,64665462,
%T A369671 333763444,1776226471,9670530120,53128162973,291546645940,
%U A369671 1592977754671,8685610041084,47462008167381,260789472093044,1442162566738036,8016343531922084,44697615509640615,249596790724248848
%N A369671 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = theta_4(x).
%C A369671 Note: theta_4(x) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2) - see A002448.
%C A369671 Congruences:
%C A369671 (C.1) a(2*n) == 0 (mod 4) for n >= 1.
%C A369671 (C.2) a(n) == A369672(n) (mod 4) for n >= 1.
%C A369671 (C.3) a(2*n)/4 == -A369672(2*n)/4 (mod 4) for n >= 1.
%H A369671 Paul D. Hanna, <a href="/A369671/b369671.txt">Table of n, a(n) for n = 1..531</a>
%H A369671 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A369671 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A369671 (1) Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
%F A369671 (2) Sum_{n=-oo..+oo} x^n * (x^n + 4*A(x))^(n-1) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
%F A369671 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^n = 0.
%F A369671 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 4*x^n*A(x))^n = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
%F A369671 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 4*x^n*A(x))^(n+1) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
%F A369671 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.
%F A369671 a(n) ~ c * d^n / n^(3/2), where d = 5.9085050558... and c = 0.2952711268... - _Vaclav Kotesovec_, Feb 03 2024
%e A369671 G.f.: A(x) = x + 4*x^2 + 15*x^3 + 52*x^4 + 177*x^5 + 664*x^6 + 3038*x^7 + 16268*x^8 + 90660*x^9 + 490456*x^10 + 2541387*x^11 + 12819184*x^12 + ...
%e A369671 where Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = theta_4(x), and
%e A369671 theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 + ... + (-1)^n*2*x^(n^2) + ...
%e A369671 RELATED SERIES.
%e A369671 When we break up the doubly infinite sum into the following parts
%e A369671 P = Sum_{n>=0} (x^n - 4*A(x))^n = 1 - 3*x - 4*x^3 - 15*x^4 - 76*x^5 - 336*x^6 - 1516*x^7 - 7040*x^8 - 34403*x^9 - 175616*x^10 - 918968*x^11 - 4847040*x^12 + ...
%e A369671 N = Sum_{n>=1} x^(n^2) / (1 - 4*x^n*A(x))^n = x + 4*x^3 + 17*x^4 + 76*x^5 + 336*x^6 + 1516*x^7 + 7040*x^8 + 34401*x^9 + 175616*x^10 + 918968*x^11 + 4847040*x^12 + ...
%e A369671 we see that the sum equals P + N = theta_4(x).
%e A369671 SPECIAL VALUES.
%e A369671 (V.1) A(exp(-Pi)) = 0.05210763699884104351595933706426840151754418802521727110...
%e A369671 where Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A(exp(-Pi)))^n = (Pi/2)^(1/4)/gamma(3/4) = 0.91357913815611682140724...
%e A369671 (V.2) A(exp(-2*Pi)) = 0.001881490423764068063219673469053308038171175452456126483...
%e A369671 where Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 4*A(exp(-2*Pi)))^n = (Pi/2)^(1/4)/gamma(3/4) * 2^(1/8) = 0.99626511456090713578995...
%e A369671 (V.3) A(exp(-4*Pi)) = 0.000003487391003072013497532566545785034046098962165471423...
%e A369671 where Sum_{n=-oo..+oo} (exp(-4*n*Pi) - 4*A(exp(-4*Pi)))^n = Pi^(1/4)/gamma(3/4) * (sqrt(2) + 1)^(1/4)/2^(7/16) = 0.99999302531528758200931...
%e A369671 (V.4) A(exp(-10*Pi)) = 0.000000000000022711010683243001546817769702787327972263611...
%e A369671 where Sum_{n=-oo..+oo} (exp(-10*n*Pi) - 4*A(exp(-10*Pi)))^n = Pi^(1/4)/gamma(3/4) * 2^(7/8)/((5^(1/4) - 1)*sqrt(5*sqrt(5) + 5)) = 0.99999999999995457797863...
%o A369671 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
%o A369671 A[#A] = polcoeff( (-sum(n=-M,M, (-1)^n * x^(n^2)) + sum(n=-#A,#A, (x^n - 4*x*Ser(A))^n) )/4, #A); ); A[n]}
%o A369671 for(n=1,30,print1(a(n),", "))
%o A369671 (PARI) {a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4);
%o A369671 A[#A] = polcoeff( (-sum(n=-M,M, (-1)^n * x^(n^2)) + sum(n=-#A,#A, x^(n^2)/(1 - 4*x^(n+1)*Ser(A))^n) )/4, #A); ); A[n]}
%o A369671 for(n=1,30,print1(a(n),", "))
%Y A369671 Cf. A369672 (dual), A002448 (theta_4), A355868.
%K A369671 nonn
%O A369671 1,2
%A A369671 _Paul D. Hanna_, Feb 03 2024