A244755 a(n) = Sum_{k=0..n} C(n,k) * (1 + 3^k)^(n-k).

1, 3, 13, 87, 985, 19563, 697573, 44195007, 4985202865, 987432857043, 344306650353853, 209169206074748967, 222262777197258910345, 409907753371580011362363, 1317924525238880964004945813, 7341603216747343890845790989967, 71176841502529490992224798115792225

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..90

Formula

E.g.f.: Sum_{n>=0} exp((1+3^n)*x) * x^n/n!.

O.g.f.: Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1).

a(n) ~ c * 3^(n^2/4) * 2^(n+1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/3) = EllipticTheta[3, 0, 1/3] = 1.69145968168171534134842... if n is even, and c = JacobiTheta2(0,1/3) = EllipticTheta[2, 0, 1/3] = 1.69061120307521423305296... if n is odd. - Vaclav Kotesovec, Jan 25 2015

Example

E.g.f.: A(x) = 1 + 3*x + 13*x^2/2! + 87*x^3/3! + 985*x^4/4! + 19563*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+3^0)^1 + (1+3^1)^0 = 3;
a(2) = (1+3^0)^2 + 2*(1+3^1)^1 + (1+3^2)^0 = 13;
a(3) = (1+3^0)^3 + 3*(1+3^1)^2 + 3*(1+3^2)^1 + (1+3^3)^0 = 87;
a(4) = (1+3^0)^4 + 4*(1+3^1)^3 + 6*(1+3^2)^2 + 4*(1+3^3)^1 + (1+3^4)^0 = 985; ...

Mathematica

Table[Sum[Binomial[n, k] * (1 + 3^k)^(n-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) )}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f. Sum_{n>=0} exp((1+3^n)*x)*x^n/n!" */
{a(n)=n!*polcoeff(sum(k=0, n, exp((1+3^k)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (1+3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A244754, A244756, A244760, A135079, A243918.

Sequence in context: A352121 A001831 A196561 * A002725 A324028 A373658

Adjacent sequences: A244752 A244753 A244754 * A244756 A244757 A244758

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2014

Status

approved