A107895 Euler transform of n!.

1, 1, 3, 9, 36, 168, 961, 6403, 49302, 430190, 4199279, 45326013, 535867338, 6884000262, 95453970483, 1420538043009, 22579098396600, 381704267100888, 6837775526561031, 129377310771795789, 2578101967764973314, 53965231260126083854, 1183813954026245944519

Offset: 0

Author

Thomas Wieder, May 26 2005

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..170

Crossrefs

Cf. A077365, A107894, A179327, A248871, A261047.

Programs

Maple

EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
`if`(n=0,1, add(add(d*p(d),d=numtheory[divisors](j)) *b(n-j),j=1..n)/n) end end:
A107895 := EulerTrans(n->n!):  seq(A107895(n),n=0..20);
# After Alois P. Heinz, A000335.  [Peter Luschny, Jul 07 2011]

Mathematica

EulerTrans[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; A107895 = EulerTrans[Factorial]; Table[A107895[n], {n, 0, 22}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)

Formula

a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 66/n^4 + 450/n^5 + 3679/n^6 + 35260/n^7 + 388511/n^8 + 4844584/n^9 + 67502450/n^10), for next coefficients see A248871. - Vaclav Kotesovec, Mar 14 2015

G.f.: Product_{n>=1} 1/(1-x^n)^(n!). - Vaclav Kotesovec, Aug 04 2015