A193520 a(n) = Sum_{k=0..n} G(n)/(G(k)*G(n-k)) where G(n) = Product_{k=0..n} k!.

1, 2, 4, 14, 122, 3122, 260642, 76214882, 85552669442, 381014246511362, 7442029915221081602, 632869669701185574873602, 264542347321693265938488883202, 517169258108069965039831739271321602, 5495073385198979486456081260457854269542402

Offset: 0

Author

Paul D. Hanna, Jul 29 2011

Keywords

• nonn

Examples

			Let F(x) = 1 + x + x^2/(1!*2!) + x^3/(1!*2!*3!) + x^4/(1!*2!*3!*4!) +...+ x^n/G(n) +...
then
F(x)^2 = 1 + 2*x + 4*x^2/(1!*2!) + 14*x^3/(1!*2!*3!) + 122*x^4/(1!*2!*3!*4!) + 3122*x^5/(1!*2!*3!*4!*5!) +...+ a(n)*x^n/G(n) +...
Illustration of initial terms:
a(3) = 1 + 3! + 3! + 1 = 14;
a(4) = 1 + 4! + 4!*3!/2! + 4! + 1 = 122;
a(5) = 1 + 5! + 5!*4!/2! + 5!*4!/2! + 5! + 1 = 3122;
a(6) = 1 + 6! + 6!*5!/2! + 6!*5!*4!/(3!*2!) + 6!*5!/2! + 6! + 1 = 260642; ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A000178, A193521.

Row sums of A009963.

Programs

Magma

A009963:= func< n,k | (1/Factorial(n+1))*(&*[ Factorial(n-j+1)/Factorial(j): j in [0..k]]) >;
A193520:= func< n | (&+[A009963(n,j): j in [0..n]]) >;
[A193520(n): n in [0..20]]; // G. C. Greubel, Jan 05 2022

Mathematica

Table[Sum[BarnesG[n+2] / (BarnesG[k+2] * BarnesG[n-k+2]), {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Mar 04 2019 *)

PARI

{a(n)=sum(k=0,n,prod(j=0,n,j!)/(prod(j=0,k,j!)*prod(j=0,n-k,j!)))}

PARI

{a(n)=prod(k=1,n,k!)*polcoeff((sum(m=0,n+1,x^m/prod(k=0,m,k!)+x*O(x^n))^2),n)}

Sage

from mpmath import mp
mp.dps = 98; mp.pretty = True
def superbinomial(n,k):
    return mp.superfac(n)/(mp.superfac(k)*mp.superfac(n-k))
def A193520(n): return add(superbinomial(n,k) for k in (0..n))
[int(A193520(n)) for n in (0..14)]  # Peter Luschny, Nov 28 2012

Formula

G.f.: A(x) = ( Sum_{n>=0} x^n/G(n) )^2 where A(x) = Sum_{n>=0} a(n)*x^n/G(n), and G(n) = Product_{k=0..n} k!.

a(n) ~ 2^(n^2/4 + n - 5*(-1)^n/8 + 23/24) * n^(n^2/4 + (-1)^n/8 - 13/24) / (sqrt(Pi) * exp(3*n^2/8 + Zeta'(-1))). - Vaclav Kotesovec, Mar 04 2019