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A370025
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Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*A(x))^n = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).
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12
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1, 5, 28, 169, 1054, 6667, 42627, 275211, 1791132, 11731613, 77242391, 510826889, 3391115560, 22586150402, 150866419771, 1010290295683, 6780795305121, 45602955247738, 307252705965207, 2073546683753911, 14014659243408481, 94851805738129599, 642767262413178788, 4360774590348465669
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OFFSET
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1,2
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COMMENTS
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A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
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LINKS
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FORMULA
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G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*A(x))^n = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 5*A(x))^(n-1) = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 5*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^n)^n = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^n)^(n+1) = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 5*A(x)*x^n)^(n+1) = 0.
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EXAMPLE
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G.f.: A(x) = x + 5*x^2 + 28*x^3 + 169*x^4 + 1054*x^5 + 6667*x^6 + 42627*x^7 + 275211*x^8 + 1791132*x^9 + 11731613*x^10 + 77242391*x^11 + 510826889*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*A(x))^n = 1 - 7*x + 7*x^4 - 7*x^9 + 7*x^16 - 7*x^25 + 7*x^36 - 7*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05561899448885311185126383683351896617798185829954077412...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 5*A)^n = (7*(Pi/2)^(1/4)/gamma(3/4) - 5)/2 = 0.69752698354640887492534...
(V.2) Let A = A(exp(-2*Pi)) = 0.001885063870555508038278982205994616246272805466135524875...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 5*A)^n = (7*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5)/2 = 0.986927900963174975264...
(V.3) Let A = A(-exp(-Pi)) = -0.03567173485605183837843763169616623725553901880108539739...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 5*A)^n = (7*Pi^(1/4)/gamma(3/4) - 5)/2 = 1.302521839246578051013...
(V.4) Let A = A(-exp(-2*Pi)) = -0.0018536159060139689998658447922411419770684746756327600438...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 5*A)^n = (7*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5)/2 = 1.01307209920708681866...
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PROG
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(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 5*Ser(A))^m ) - 1 - 7*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/5 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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