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A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.

%I A309972 #22 Dec 07 2020 13:59:44
%S A309972 1,1,2,18,6912,216000000,1632586752000000000,
%T A309972 498266101635303733401600000000000,
%U A309972 1140494258799407218656986754465090350453096448000000000000000
%N A309972 Product of multinomial coefficients M(n;lambda), where lambda ranges over all partitions of n.
%H A309972 Alois P. Heinz, <a href="/A309972/b309972.txt">Table of n, a(n) for n = 0..14</a>
%H A309972 Wikipedia, <a href="https://en.wikipedia.org/wiki/Multinomial_theorem#Multinomial_coefficients">Multinomial coefficients</a>
%H A309972 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A309972 a(n) = Product_{k=1..A000041(n)} A036038(n,k).
%F A309972 a(n) = A309951(n,A000041(n)).
%e A309972 a(3) = M(3;3) * M(3;2,1) * M(3;1,1,1) = 1 * 3 * 6 = 18.
%p A309972 b:= proc(n, i) option remember; `if`(n=0 or i=1, [n!], [map(t->
%p A309972       binomial(n, i)*t, b(n-i, min(n-i, i)))[], b(n, i-1)[]])
%p A309972     end:
%p A309972 a:= n-> mul(i, i=b(n$2)):
%p A309972 seq(a(n), n=0..9);  # _Alois P. Heinz_, Aug 25 2019
%t A309972 b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {n!}, Join[Binomial[n, i] #& /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
%t A309972 a[n_] := Times @@ b[n, n];
%t A309972 a /@ Range[0, 9] (* _Jean-François Alcover_, Dec 07 2020, after _Alois P. Heinz_ *)
%Y A309972 Rightmost terms in rows of A309951.
%Y A309972 Cf. A000041, A005651, A036038, A078760, A210237.
%K A309972 nonn
%O A309972 0,3
%A A309972 _Alois P. Heinz_, Aug 25 2019