OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..89
FORMULA
E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * (2*x)^n/n!.
O.g.f.: Sum_{n>=0} (2*x)^n/(1 - (1+3^n)*x)^(n+1).
a(n) ~ c * 3^(n^2/4) * 2^((3*n+1)/2) / sqrt(Pi*n), where c = sum_{k=-inf..+inf} 1/(3^(k^2) * 2^k) = 1.88621563508001862566062... if n is even, and c = sum_{k=-inf..+inf} 1/(3^((k+1/2)^2) * 2^(k+1/2)) = 1.88659407336643412717014... if n is odd. - Vaclav Kotesovec, Jan 25 2015
EXAMPLE
E.g.f.: A(x) = 1 + 4*x + 24*x^2/2! + 232*x^3/3! + 3840*x^4/4! + 111904*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+3^0)^1 + (1+3^1)^0*2 = 4;
a(2) = (1+3^0)^2 + 2*(1+3^1)^1*2 + (1+3^2)^0*2^2 = 24;
a(3) = (1+3^0)^3 + 3*(1+3^1)^2*2 + 3*(1+3^2)^1*2^2 + (1+3^3)^0*2^3 = 232;
a(4) = (1+3^0)^4 + 4*(1+3^1)^3*2 + 6*(1+3^2)^2*2^2 + 4*(1+3^3)^1*2^3 + (1+3^4)^0*2^4 = 3480; ...
MATHEMATICA
Table[Sum[Binomial[n, k] * (1 + 3^k)^(n-k) * 2^k, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 25 2015 *)
PROG
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (1 + 3^k)^(n-k)*2^k )}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*(2*x)^n/n! */
{a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*(2*x)^k/k!), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* O.g.f. Sum_{n>=0} (2*x)^n/(1 - (1+3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, (2*x)^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 25, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 05 2014
STATUS
approved