A129481 a(n) = coefficient of x^n in n!*Product_{k=0..n} [Sum_{j=0..k} x^j/j! ].

1, 1, 3, 19, 175, 2111, 31321, 550810, 11194177, 258068893, 6653230111, 189653427206, 5922604033567, 201075967613262, 7373834652641003, 290474615891145106, 12232735359488840833, 548429151685677131389

Offset: 0

Author

Paul D. Hanna, Apr 17 2007

Keywords

• nonn

Comments

a(n) is also the number of ordered submultisets of A000707. - J. M. Bergot, Aug 13 2016

Examples

			a(2) = [x^2] 2!*(1)*(1+x)*(1+x+x^2/2!) = [x^2] (2 +4*x +3*x^2 +x^3) = 3.
a(3) = [x^3] 3!*(1)*(1+x)*(1 + x + x^2/2!)*(1 + x + x^2/2! + x^3/3!) =
[x^3] (6 + 18*x + 24*x^2 + 19*x^3 +...) = 19.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..150

Crossrefs

Cf. A000707.

Programs

Magma

m:=30; R:=PowerSeriesRing(Integers(), m+2);
p:= func< n,x | (&*[ (&+[x^j/Factorial(j): j in [0..k]]) : k in [0..n]]) >;
A129481:= func< n | Coefficient(R!(Laplace( p(n,x) )), n) >;
[A129481(n): n in [0..m]]; // G. C. Greubel, Feb 12 2024

Mathematica

Flatten[{1,Table[Coefficient[Expand[n!*Product[Sum[x^j/j!,{j,0,k}],{k,0,n}]],x^n],{n,1,20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

PARI

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

SageMath

def p(n,x): return product(sum(x^j/factorial(j) for j in range(k+1)) for k in range(n+1))
def A129481(n): return factorial(n)*( p(n,x) ).series(x, 101).list()[n]
[A129481(n) for n in range(31)] # G. C. Greubel, Feb 13 2024

Formula

a(n) ~ c * n^n, where c = 0.660942456683588459181273625114230472913... . - Vaclav Kotesovec, Feb 10 2015