A158876 Expansion of e.g.f.: exp( Sum_{n>=1} (n-1)! * x^n ).

1, 1, 3, 19, 217, 4041, 113611, 4532683, 244208049, 17085010897, 1504881245971, 162835665686211, 21219897528855433, 3276502399914104089, 591351260856215820507, 123322423833602768272891, 29423834155886520870184801, 7963056392690313008566254753

Offset: 0

Author

Paul D. Hanna, Apr 13 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 19*x^3/3! + 217*x^4/4! +...
log(A(x)) = x + x^2 + 2!*x^3 + 3!*x^4 +...+ (n-1)!*x^n +....
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..253

Crossrefs

Cf. A000142, A122949.

Programs

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( Exp( &+[Factorial(j-1)*x^j: j in [1..m+2]] ) ))); // G. C. Greubel, Mar 04 2020

Maple

m:=20; S:=series( exp(add((j-1)!*x^j, j=1..m+2)), x, m+1): seq(j!*coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 04 2020

Mathematica

With[{m = 20}, CoefficientList[Series[Exp[Sum[(j-1)!*x^j, {j, m+2}]], {x, 0, m}], x]*Range[0, m]!] (* G. C. Greubel, Mar 04 2020 *)

PARI

{a(n)=if(n==0,1,(n-1)!*sum(k=1,n,k!*a(n-k)/(n-k)!))}
for(n=0,20,print1(a(n),", "))

PARI

{a(n)=n!*polcoeff(exp(sum(k=1,n,(k-1)!*x^k)+x*O(x^n)),n)}
for(n=0,20,print1(a(n),", "))

Sage

m=20
def A158876_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp(sum(factorial(j-1)*x^j for j in (1..m+2))) ).list()
a=A158876_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Mar 04 2020

Formula

a(n) = (n-1)! * Sum_{k=1..n} k! * a(n-k) / (n-k)! for n>0 with a(0)=1.

E.g.f. A(x) satisfies:

(1) A'(x)/A(x) = Sum_{k>=0} (n+1)! * x^n.

(2) A(x) = exp(x + x^2 * A'(x)/A(x)).

Let F(x) = Sum_{n>=0} n! * x^n, then

(3) [x^n] A(x)^n * (2 - F(x)) = 0 for n > 0,

(4) [x^n] A(x) * (n + 1 - F(x)) = 0 for n > 0. - Paul D. Hanna, May 26 2018

a(n) ~ n! * (n-1)!. - Vaclav Kotesovec, Aug 01 2017