OFFSET
0,1
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..330
FORMULA
G.f.: Sum_{n>=0} (1+x)^n/2^n * Product_{k=1..n} (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)) due to a q-series identity.
G.f.: 1/(1 - (1+x)/2 /(1 - (1+x)*((1+x)^2-1)/2 /(1 - (1+x)^5/2 /(1 - (1+x)^3*((1+x)^4-1)/2 /(1 - (1+x)^9/2 /(1 - (1+x)^5*((1+x)^6-1)/2 /(1 - (1+x)^13/2 /(1 - (1+x)^7*((1+x)^8-1)/2 /(1 - ...))))))))), a continued fraction due to a partial elliptic theta function identity.
a(n) = Sum_{k>=sqrt(n)} binomial(k^2,n) / 2^k.
a(n) = Sum_{k=0..2*n} A303920(n,k) * 2^k, for n>0.
a(n) = 2 * A173217(n) for n>=0.
a(n) ~ 2^(2*n + 1/2 - log(2)/8) * n^n / (exp(n) * log(2)^(2*n + 1)). - Vaclav Kotesovec, Oct 08 2019
EXAMPLE
G.f.: A(x) = 2 + 6*x + 72*x^2 + 1488*x^3 + 43212*x^4 + 1615824*x^5 + 73897824*x^6 + 3995603040*x^7 + 249332628600*x^8 + 17635891224600*x^9 +...
where
A(x) = 1 + (1+x)/2 + (1+x)^4/2^2 + (1+x)^9/2^3 + (1+x)^16/2^4 + (1+x)^25/2^5 + (1+x)^36/2^6 + (1+x)^49/2^7 + (1+x)^64/2^8 +...+ (1+x)^(n^2)/2^n +...
MATHEMATICA
Table[Round[Sum[Binomial[k^2, n]/2^k, {k, Sqrt[n], Infinity}]] , {n, 0, 20}] (* G. C. Greubel, May 23 2017 *)
Table[2*Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -2*j, 0]/2, {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Oct 08 2019 *)
PROG
(PARI) /* Informal listing of terms: */
{Vec( round( sum(n=0, 600, (1+x +O(x^31))^(n^2)/2^n * 1.) ) )}
{Vec( round( sum(n=0, 200, (1.+x)^n/2^n * prod(k=1, n, (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)) +O(x^21) ) ) ) )}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 23 2015
STATUS
approved