[go: up one dir, main page]

login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

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