A244754 a(n) = Sum_{k=0..n} C(n,k) * (1 + 2^k)^(n-k).

1, 3, 11, 51, 311, 2583, 30011, 495771, 11740271, 399511023, 19531952051, 1369534859091, 137461591250951, 19708614005005383, 4029559971566918891, 1172950335844577723211, 485515762655939377001951, 285459356061242116657495263, 238215406681004045293498284131

Offset

0,2

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

E.g.f.: Sum_{n>=0} exp((1+2^n)*x) * x^n/n!.

O.g.f.: Sum_{n>=0} x^n/(1 - (1+2^n)*x)^(n+1).

a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0, 1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586... if n is even, and c = JacobiTheta2(0, 1/2) = EllipticTheta[2, 0, 1/2] = 2.12893125051302755859... if n is odd. - Vaclav Kotesovec, Jan 25 2015

Example

E.g.f.: A(x) = 1 + 3*x + 11*x^2/2! + 51*x^3/3! + 311*x^4/4! + 2583*x^5/5! +...
where
A(x) = exp(x)*(1 + 2*x + 6*x^2/2! + 26*x^3/3! + 162*x^4/4! + 1442*x^5/5! +...+ A047863(n)*x^n/n! +...).
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+2^0)^1 + (1+2^1)^0 = 3;
a(2) = (1+2^0)^2 + 2*(1+2^1)^1 + (1+2^2)^0 = 11;
a(3) = (1+2^0)^3 + 3*(1+2^1)^2 + 3*(1+2^2)^1 + (1+2^3)^0 = 51;
a(4) = (1+2^0)^4 + 4*(1+2^1)^3 + 6*(1+2^2)^2 + 4*(1+2^3)^1 + (1+2^4)^0 = 311; ...

Mathematica

Table[Sum[Binomial[n, k] * (1 + 2^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 + 2^k)^(n-k) )}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* E.g.f.: Sum_{n>=0} exp((1+2^n)*x)*x^n/n!: */
{a(n)=n!*polcoeff(sum(k=0, n, exp((1+2^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+2^n)*x)^(n+1): */
{a(n)=polcoeff(sum(k=0, n, x^k/(1-(1+2^k)*x +x*O(x^n))^(k+1)), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A047863, A244755, A243918.

Sequence in context: A230008 A007047 A182176 * A129097 A319155 A362468

Adjacent sequences: A244751 A244752 A244753 * A244755 A244756 A244757

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2014

Status

approved