A244689 a(n) = Sum_{k=0..n} C(n,k) * (n + 3*2^k)^(n-k) * 2^(k^2).

1, 6, 73, 1934, 157857, 56192650, 92426525425, 666550826226318, 20291280723841291105, 2550027209175411070031954, 1305537190872353152721812616649, 2701765523097192231845112449534664934, 22497928378023184347083511140879821373194561, 751862888756012808502475142804126477229231539927258

Offset

0,2

Links

Table of n, a(n) for n=0..13.

Formula

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

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 11 2014

Example

E.g.f.: A(x) = 1 + 6*x + 73*x^2/2! + 1934*x^3/3! + 157857*x^4/4! + 56192650*x^5/5! +...
ILLUSTRATION OF INITIAL TERMS:
a(1) = (1+3*2^0)^1*2^0 + (1+3*2^1)^0*2^1 = 6;
a(2) = (2+3*2^0)^2*2^0 + 2*(2+3*2^1)^1*2^1 + (2+3*2^2)^0*2^4 = 73;
a(3) = (3+3*2^0)^3*2^0 + 3*(3+3*2^1)^2*2^1 + 3*(3+3*2^2)^1*2^4 + (3+3*2^3)^0*2^9 = 1934;
a(4) = (4+3*2^0)^4*2^0 + 4*(4+3*2^1)^3*2^1 + 6*(4+3*2^2)^2*2^4 + 4*(4+3*2^3)^1*2^9 + (4+3*2^4)^0*2^16 = 157857; ...
where we have the binomial identity:
a(1) = 1^1*(3+2^0)^0 + 1^1*(3+2^1)^1 = 6;
a(2) = 2^2*(3+2^0)^0 + 2*2^1*(3+2^1)^1 + 2^0*(3+2^2)^2 = 73;
a(3) = 3^3*(3+2^0)^0 + 3*3^2*(3+2^1)^1 + 3*3^1*(3+2^2)^2 + 3^0*(3+2^3)^3 = 1934;
a(4) = 4^4*(3+2^0)^0 + 4*4^3*(3+2^1)^1 + 6*4^2*(3+2^2)^2 + 4*4^1*(3+2^3)^3 + 4^0*(3+2^4)^4 = 157857; ...

Mathematica

Table[Sum[Binomial[n, k] * (n + 3*2^k)^(n-k) * 2^(k^2), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 11 2014 *)

Prog

(PARI) {a(n) = sum(k=0, n, binomial(n, k) * (n + 3*2^k)^(n-k) * 2^(k^2) )}
for(n=0, 15, print1(a(n), ", "))
(PARI) {a(n) = sum(k=0, n, binomial(n, k) * n^(n-k) * (3 + 2^k)^k )}
for(n=0, 15, print1(a(n), ", "))

Crossrefs

Cf. A244751.

Sequence in context: A135594 A346960 A168603 * A058793 A066171 A269647

Adjacent sequences: A244686 A244687 A244688 * A244690 A244691 A244692

Keyword

nonn

Author

Paul D. Hanna, Jul 05 2014

Status

approved