A123681 a(n) = (1/(n+1)) * Sum_{k=0..n} C(n+k-1,k)*k! = A123680(n)/(n+1).

1, 1, 3, 19, 197, 2841, 52327, 1171871, 30899529, 937529317, 32173291931, 1232093935227, 52088478142861, 2409578607253169, 121067200114483407, 6565538372492694871, 382234458749760846737, 23777755561583494209981

Offset: 0

Author

Paul D. Hanna, Oct 05 2006

Keywords

• nonn

Examples

			a(n) = (Pochhammer(n, n + 1)*subfactorial(-2*n - 1) + (-1)^n*subfactorial(-n))/(n+1) where subfactorial(n) = exp(-1)*Gamma(n + 1, -1). - _Peter Luschny_, Oct 18 2017
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..365

Crossrefs

Cf. A123680.

Programs

Maple

subfactorial := n -> simplify(exp(-1)*GAMMA(n+1,-1)):
a := n -> (pochhammer(n,n+1)*subfactorial(-2*n-1)+(-1)^n*subfactorial(-n))/(n+1):
seq(simplify(evalc(a(n))), n=0..17); # Peter Luschny, Oct 18 2017

Mathematica

Table[1/(n+1) Sum[Binomial[n+k-1,k]k!,{k,0,n}],{n,0,20}] (* Harvey P. Dale, Dec 14 2012 *)

PARI

a(n)=sum(k=0,n,binomial(n+k-1,k)*k!)/(n+1)

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} k! * [x^k] 1/(1-x)^n.

a(n) ~ 2^(2*n - 1/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 27 2017

Extensions

Definition corrected by Harvey P. Dale, Dec 14 2012