A138903 a(n) = (1/2^n)* Sum_{k=0..n} binomial(n,k)*(n+k)^(n-1).

1, 3, 21, 234, 3590, 70254, 1672972, 46955760, 1517994792, 55549351800, 2269918543640, 102452561694864, 5062050729973120, 271751784988056576, 15750949414628405760, 980315266648197537792, 65207656047198387921536

Offset: 1

Author

Paul D. Hanna and Vladeta Jovovic, Apr 02 2008, Apr 03 2008

Keywords

• easy
• nonn

Programs

Maple

A138903 := proc(n) local k ; add(binomial(n,k)*(n+k)^(n-1),k=0..n)/2^n ; end: seq(A138903(n),n=1..20) ; # R. J. Mathar, Apr 12 2008

Mathematica

Table[1/2^n * Sum[Binomial[n,k]*(n+k)^(n-1),{k,0,n}],{n,1,20}] (* Vaclav Kotesovec, Jun 15 2013 *)

PARI

{a(n)=local(X=x+x*O(x^n)); n!*polcoeff(serreverse(2*x/(exp(X)+exp(2*X)) ), n)}

Formula

E.g.f.: A(x) = log(B(x)), where B(x) is e.g.f. of A138860.

E.g.f.: A(x) = Series_Reversion[ 2*x/(exp(x) + exp(2*x)) ].

a(n) ~ n^(n-1)*(1+r)^n*r^n/(sqrt(1+3*r)*(1-r)^(2*n)*exp(n)*2^n), where r = 0.6472709258412625... is the root of the equation (r/(1-r))^(1+r) = e. - Vaclav Kotesovec, Jun 15 2013

Extensions

More terms from R. J. Mathar, Apr 12 2008