OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..253
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
EXAMPLE
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 +....
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 *)
PROG
(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), ", "))
(Magma) m:=20; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Laplace( Exp( &+[Factorial(j-1)*x^j: j in [1..m+2]] ) ))); // G. C. Greubel, Mar 04 2020
(Sage)
m=20
def A158876_list(prec):
P.<x> = 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 13 2009
STATUS
approved