A369671 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = theta_4(x).
1, 4, 15, 52, 177, 664, 3038, 16268, 90660, 490456, 2541387, 12819184, 64665462, 333763444, 1776226471, 9670530120, 53128162973, 291546645940, 1592977754671, 8685610041084, 47462008167381, 260789472093044, 1442162566738036, 8016343531922084, 44697615509640615, 249596790724248848
Offset: 1
Keywords
Examples
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 + ...
where Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = theta_4(x), and
theta_4(x) = 1 - 2*x + 2*x^4 - 2*x^9 + 2*x^16 + ... + (-1)^n*2*x^(n^2) + ...
RELATED SERIES.
When we break up the doubly infinite sum into the following parts
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 + ...
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 + ...
we see that the sum equals P + N = theta_4(x).
SPECIAL VALUES.
(V.1) A(exp(-Pi)) = 0.05210763699884104351595933706426840151754418802521727110...
where Sum_{n=-oo..+oo} (exp(-n*Pi) - 4*A(exp(-Pi)))^n = (Pi/2)^(1/4)/gamma(3/4) = 0.91357913815611682140724...
(V.2) A(exp(-2*Pi)) = 0.001881490423764068063219673469053308038171175452456126483...
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...
(V.3) A(exp(-4*Pi)) = 0.000003487391003072013497532566545785034046098962165471423...
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...
(V.4) A(exp(-10*Pi)) = 0.000000000000022711010683243001546817769702787327972263611...
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...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..531
- Eric Weisstein's World of Mathematics, Jacobi Theta Functions
Programs
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PARI
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4); 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]} for(n=1,30,print1(a(n),", ")) -
PARI
{a(n) = my(A=[1]); for(i=1,n, A = concat(A,0); M=sqrtint(#A+4); 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]} for(n=1,30,print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (x^n - 4*A(x))^n = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + 4*A(x))^(n-1) = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 4*x^n*A(x))^n = Sum_{n=-oo..+oo} (-1)^n * x^(n^2).
(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).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 4*x^n*A(x))^n = 0.
a(n) ~ c * d^n / n^(3/2), where d = 5.9085050558... and c = 0.2952711268... - Vaclav Kotesovec, Feb 03 2024
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