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A107894 Sum over the products of factorials of parts in all partitions of n where the sum runs over the number of different parts only.

%I A107894 #20 Oct 29 2015 07:09:12
%S A107894 1,1,3,9,35,167,943,6379,48945,429651,4189865,45307601,535518109,
%T A107894 6883110373,95435065935,1420468921893,22577620176887,381695573051099,
%U A107894 6837601709298811,129375694813679215,2578070946813526485,53964818587883937807,1183805926540690127573
%N A107894 Sum over the products of factorials of parts in all partitions of n where the sum runs over the number of different parts only.
%H A107894 Alois P. Heinz, <a href="/A107894/b107894.txt">Table of n, a(n) for n = 0..200</a>
%F A107894 a(n) ~ n! * (1 + 1/n + 3/n^2 + 12/n^3 + 65/n^4 + 443/n^5 + 3626/n^6 + 34811/n^7 + 384479/n^8 + 4806098/n^9 + 67109281/n^10), for coefficients see A256124. - _Vaclav Kotesovec_, Mar 15 2015
%e A107894 The partitions of 5 are 1+1+1+1+1, 1+1+1+2, 1+1+3, 1+2+2, 1+4, 2+3, 5, the corresponding products of factorials of parts are (when multiple parts are counted once only) 1!, 1!*2!, 1!*3!, 1!*2!, 1!*4!, 2!*3!, 5! and their sum is a(5) = 167.
%p A107894 b:= proc(n, i) option remember;
%p A107894       `if`(n=0 or i<2, 1, b(n, i-1) +i!*add(b(n-i*j, i-1), j=1..n/i))
%p A107894     end:
%p A107894 a:= n-> b(n, n):
%p A107894 seq(a(n), n=0..30); # _Alois P. Heinz_, Apr 04 2012
%t A107894 Total[Times@@@(Union/@IntegerPartitions[#]!)]&/@Range[20]  (* _Harvey P. Dale_, Feb 26 2011 *)
%t A107894 b[n_, i_] := b[n, i] = If[n==0 || i<2, 1, b[n, i-1] + i!*Sum[b[n-i*j, i-1], {j, 1, n/i}]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Oct 29 2015, after _Alois P. Heinz_ *)
%Y A107894 Cf. A077365, A107895, A256124.
%K A107894 nonn
%O A107894 0,3
%A A107894 _Thomas Wieder_, May 26 2005
%E A107894 a(0) inserted and more terms from _Alois P. Heinz_, Apr 04 2012