A251661 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^(n-k).

1, 3, 15, 123, 1671, 37863, 1447515, 93965763, 10456301871, 2001375249423, 663553617119475, 381265346343864843, 381607689867265672551, 664239239404717367975223, 2018751151993358704057734795, 10680818706115450217386068210963, 98710608829560784063971722066895711

Offset

0,2

Comments

a(n) == 0 (mod 3) for n>0. - Paul D. Hanna, May 02 2015

Links

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

Formula

E.g.f.: Sum_{n>=0} exp((2^n + 3^n)*x) * x^n/n!.

O.g.f.: Sum_{n>=0} x^n/(1 - (2^n + 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 + 15*x^2/2! + 123*x^3/3! + 1671*x^4/4! + 37863*x^5/5! +...
where
A(x) = exp(2*x) + exp(5*x)*x + exp(13*x)*x^2/2! + exp(35*x)*x^3/3! + exp(97*x)*x^4/4! + exp(275*x)*x^5/5! + exp(793*x)*x^6/6! +...+ exp((2^n+3^n)*x)*x^n/n! +...
ILLUSTRATION OF INITIAL TERMS:
a(0) = 1*(2^0+3^0)^0 = 1;
a(1) = 1*(2^0+3^0)^1 + 1*(2^1+3^1)^0 = 3;
a(2) = 1*(2^0+3^0)^2 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^0 = 15;
a(3) = 1*(2^0+3^0)^3 + 3*(2^1+3^1)^2 + 3*(2^2+3^2)^1 + 1*(2^3+3^3)^0 = 123;
a(4) = 1*(2^0+3^0)^4 + 4*(2^1+3^1)^3 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^1 + 1*(2^4+3^4)^0 = 1671; ...

Mathematica

Table[Sum[Binomial[n, k] * (2^k + 3^k)^(n-k), {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jan 25 2015 *)

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (2^k + 3^k)^(n-k) )}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} exp((2^n + 3^n)*x)*x^n/n!: */
{a(n)=n!*polcoeff(sum(k=0, n, exp((2^k + 3^k)*x +x*O(x^n))*x^k/k!), n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) /* O.g.f. Sum_{n>=0} x^n/(1 - (2^n + 3^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k + 3^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A244754, A251671.

Sequence in context: A348903 A191371 A113077 * A230657 A246573 A270503

Adjacent sequences: A251658 A251659 A251660 * A251662 A251663 A251664

Keyword

nonn

Author

Paul D. Hanna, Jan 21 2015

Status

approved