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A325504 Product of products of parts over all strict integer partitions of n.

%I A325504 #14 Feb 23 2024 20:31:26
%S A325504 1,1,2,6,12,120,1440,40320,1209600,1567641600,2633637888000,
%T A325504 13905608048640000,5046067048690483200000,
%U A325504 5289893008483207348224000000,1266933607446134946465526579200000000,99304891373531545064656621572980736000000000000
%N A325504 Product of products of parts over all strict integer partitions of n.
%H A325504 Alois P. Heinz, <a href="/A325504/b325504.txt">Table of n, a(n) for n = 0..30</a>
%F A325504 A001222(a(n)) = A325515(n).
%F A325504 a(n) = A003963(A325506(n)).
%e A325504 The strict partitions of 5 are {(5), (4,1), (3,2)} with product a(5) = 5*4*1*3*2 = 120.
%e A325504 The sequence of terms together with their prime indices begins:
%e A325504               1: {}
%e A325504               1: {}
%e A325504               2: {1}
%e A325504               6: {1,2}
%e A325504              12: {1,1,2}
%e A325504             120: {1,1,1,2,3}
%e A325504            1440: {1,1,1,1,1,2,2,3}
%e A325504           40320: {1,1,1,1,1,1,1,2,2,3,4}
%e A325504         1209600: {1,1,1,1,1,1,1,1,2,2,2,3,3,4}
%e A325504      1567641600: {1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,4}
%e A325504   2633637888000: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,4,4}
%p A325504 a:= n-> mul(i, i=map(x-> x[], select(x->
%p A325504         nops(x)=nops({x[]}), combinat[partition](n)))):
%p A325504 seq(a(n), n=0..15);  # _Alois P. Heinz_, Aug 03 2021
%p A325504 # second Maple program:
%p A325504 b:= proc(n, i) option remember; `if`(n=0, [1$2], `if`(i<1, [0, 1], ((f, g)->
%p A325504      [f[1]+g[1], f[2]*g[2]*i^g[1]])(b(n, i-1), b(n-i, min(n-i, i-1)))))
%p A325504     end:
%p A325504 a:= n-> b(n$2)[2]:
%p A325504 seq(a(n), n=0..15);  # _Alois P. Heinz_, Aug 03 2021
%t A325504 Table[Times@@Join@@Select[IntegerPartitions[n],UnsameQ@@#&],{n,0,10}]
%Y A325504 Cf. A000009, A006128, A007870 (non-strict version), A015723, A022629 (sum of products of parts), A066186, A066189, A066633, A246867, A325505, A325506, A325512, A325513, A325515.
%K A325504 nonn
%O A325504 0,3
%A A325504 _Gus Wiseman_, May 07 2019