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A160564 Sum of products of factorials of parts times the factorial of the number of parts in all integer partitions of n.

%I A160564 #11 Sep 03 2023 21:04:11
%S A160564 1,1,4,16,80,420,2592,17352,132240,1117200,10559040,110276352,
%T A160564 1268640000,15923168640,216767367936,3178157607936,49918919122944,
%U A160564 835744605027840,14852897362759680,279172076525153280,5531978038112409600,115241366146485749760
%N A160564 Sum of products of factorials of parts times the factorial of the number of parts in all integer partitions of n.
%C A160564 Take each Ferrers diagram of the partitions of n, label the cells within each row and then linearly order the rows.
%H A160564 Alois P. Heinz, <a href="/A160564/b160564.txt">Table of n, a(n) for n = 0..450</a>
%e A160564 a(3) = 16 because the partitions of 3 can be so ordered in 16 ways: 3 (6); 2,1 (4); 1,1,1 (6).
%p A160564 b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,
%p A160564       add(b(n-i*j, i-1, p+j)*i!^j, j=0..n/i)))
%p A160564     end:
%p A160564 a:= n-> b(n$2, 0):
%p A160564 seq(a(n), n=0..23);  # _Alois P. Heinz_, Oct 02 2017
%t A160564 p = Table[Map[Function[n, Apply[Times, n! ]], Partitions[i]], {i, 0, 20}]; q = Table[Map[Function[n, Length[n]! ], Partitions[i]], {i, 0, 20}]; Map[Function[n, Apply[Plus, n]], p*q]
%Y A160564 Cf. A101880, A077365, A126787.
%K A160564 nonn
%O A160564 0,3
%A A160564 _Geoffrey Critzer_, May 19 2009