A123680 - OEIS

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

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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 *)