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%I #24 Aug 08 2015 08:02:55
%S 1,1,3,9,36,168,961,6403,49302,430190,4199279,45326013,535867338,
%T 6884000262,95453970483,1420538043009,22579098396600,381704267100888,
%U 6837775526561031,129377310771795789,2578101967764973314,53965231260126083854,1183813954026245944519
%N Euler transform of n!.
%H Alois P. Heinz, <a href="/A107895/b107895.txt">Table of n, a(n) for n = 0..170</a>
%F 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
%F G.f.: Product_{n>=1} 1/(1-x^n)^(n!). - _Vaclav Kotesovec_, Aug 04 2015
%p EulerTrans := proc(p) local b; b := proc(n) option remember; local d, j;
%p `if`(n=0,1, add(add(d*p(d),d=numtheory[divisors](j)) *b(n-j),j=1..n)/n) end end:
%p A107895 := EulerTrans(n->n!): seq(A107895(n),n=0..20);
%p # After _Alois P. Heinz_, A000335. [_Peter Luschny_, Jul 07 2011]
%t 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_ *)
%Y Cf. A077365, A107894, A179327, A248871, A261047.
%K nonn
%O 0,3
%A _Thomas Wieder_, May 26 2005
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