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

1, 2, 9, 76, 985, 17046, 366289, 9374968, 278095761, 9375293170, 353906211241, 14785127222724, 677150215857193, 33734100501544366, 1816008001717251105, 105048613959883117936, 6497985798745934394529, 427999600108502895779658

Offset

0,2

Comments

a(n) is divisible by (n+1): a(n)/(n+1) = A123681(n).

Links

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

Formula

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

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

Example

Since a(n) = Sum_{k=0..n} k! * [x^k] 1/(1-x)^n, to get a(4),
list coefficients of x^0 through x^4 in 1/(1-x)^4, [1,4,10,20,35],
then dot product with factorials 0! through 4!, [0!,1!,2!,3!,4! ],
so that a(4) = 1*0! + 4*1! + 10*2! + 20*3! + 35*4! = 985.

Maple

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

Mathematica

Table[Sum[Binomial[n + k - 1, k]*k!, {k, 0, n}], {n, 0, 20}] (* G. C. Greubel, Oct 18 2017 *)

Prog

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

Crossrefs

Cf. A123681.

Sequence in context: A245406 A337558 A276742 * A132621 A108992 A058164

Adjacent sequences: A123677 A123678 A123679 * A123681 A123682 A123683

Keyword

nonn

Author

Paul D. Hanna, Oct 05 2006

Extensions

Definition corrected Oct 27 2006

Status

approved