A244753 a(n) = Sum_{k=0..n} C(n,k) * (n + 2^k)^k.

1, 4, 45, 1494, 167321, 70339178, 118127981277, 818113700595166, 23602509419592675345, 2828610325004443707717522, 1397057668479359172475738134221, 2819825298850525709434498781240666534, 23097546959835633409694123795378169234099369

Offset

0,2

Links

Table of n, a(n) for n=0..12.

Formula

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

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Aug 24 2017

Example

E.g.f.: A(x) = 1 + 4*x + 45*x^2/2! + 1494*x^3/3! + 167321*x^4/4! + 70339178*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+2^0)^0 + (1+2^1)^1 = 4;
a(2) = (2+2^0)^0 + 2*(2+2^1)^1 + (2+2^2)^2 = 45;
a(3) = (3+2^0)^0 + 3*(3+2^1)^1 + 3*(3+2^2)^2 + (3+2^3)^3 = 1494;
a(4) = (4+2^0)^0 + 4*(4+2^1)^1 + 6*(4+2^2)^2 + 4*(4+2^3)^3 + (4+2^4)^4 = 167321; ...
where we have the binomial identity:
a(1) = (1+1*2^0)^1*2^0 + (1+2^1)^0*2^1 = 4;
a(2) = (1+2*2^0)^2*2^0 + 2*(1+2*2^1)^1*2^1 + (1+2*2^2)^0*2^4 = 45;
a(3) = (1+3*2^0)^3*2^0 + 3*(1+3*2^1)^2*2^1 + 3*(1+3*2^2)^1*2^4 + (1+3*2^3)^0*2^9 = 1494;
a(4) = (1+4*2^0)^4*2^0 + 4*(1+4*2^1)^3*2^1 + 6*(1+4*2^2)^2*2^4 + 4*(1+4*2^3)^1*2^9 + (1+4*2^4)^0*2^16 = 167321; ...

Mathematica

Table[Sum[Binomial[n, k](n+2^k)^k, {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Jun 15 2017 *)

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (n + 2^k)^k )}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (1 + n*2^k)^(n-k) * 2^(k^2) )}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Sequence in context: A126452 A082765 A132873 * A335014 A364656 A102894

Adjacent sequences: A244750 A244751 A244752 * A244754 A244755 A244756

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2014

Status

approved