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A370021
Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
12
1, 1, 4, 9, 22, 63, 155, 415, 1124, 2957, 8047, 21817, 59048, 161870, 442675, 1214563, 3348145, 9228858, 25514319, 70682731, 195993889, 544578231, 1515027660, 4219560585, 11768353857, 32853953466, 91812137378, 256831830373, 719046731299, 2014808160498, 5650037329385
OFFSET
1,3
COMMENTS
A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
a(n+1)/a(n) tends to 2.874307... - Vaclav Kotesovec, Feb 11 2024
LINKS
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^(n-1) = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^n)^n = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + A(x)*x^n)^(n+1) = 1 + 3*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + x^2 + 4*x^3 + 9*x^4 + 22*x^5 + 63*x^6 + 155*x^7 + 415*x^8 + 1124*x^9 + 2957*x^10 + 8047*x^11 + 21817*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + A(x))^n = 1 - 3*x + 3*x^4 - 3*x^9 + 3*x^16 - 3*x^25 + 3*x^36 - 3*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04543932020196352081239499480519595850147996376296857684...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + A)^n = (3*(Pi/2)^(1/4)/gamma(3/4) - 1)/2 = 0.87036870723417523211086...
(V.2) Let A = A(exp(-2*Pi)) = 0.00187095623366907901234297087932572258706353074482100743...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + A)^n = (3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 1)/2 = 0.99626511456090713578995...
(V.3) Let A = A(-exp(-Pi)) = -0.04164083178192506029717066967023726841141127226704810579...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + A)^n = (3*Pi^(1/4)/gamma(3/4) - 1)/2 = 1.12965221681996202186297...
(V.4) Let A = A(-exp(-2*Pi)) = -0.00186398133004329627873834535037664668964585574963215266...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + A)^n = (3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 1)/2 = 1.00560232823160863657151...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 1*Ser(A))^m ) - 1 - 3*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1) ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved