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A328158 Number of colored integer partitions of 2n using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.

%I A328158 #10 May 08 2020 16:33:35
%S A328158 1,2,22,428,11595,416010,18283208,945843148,58252818659,4087684096527,
%T A328158 317934667075551,28164509102578546,2781331187964705790,
%U A328158 294700331738309167806,33811410232219114946609,4297801013746798965557794,593679426174377865941838598
%N A328158 Number of colored integer partitions of 2n using all colors of an n-set such that each block of part i with multiplicity j has a pattern of i*j colors in (weakly) increasing order.
%H A328158 Alois P. Heinz, <a href="/A328158/b328158.txt">Table of n, a(n) for n = 0..175</a>
%H A328158 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A328158 a(n) = A326500(2n,n).
%p A328158 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add((t->
%p A328158       b(n-t, min(n-t, i-1), k)*binomial(k+t-1, t))(i*j), j=0..n/i)))
%p A328158     end:
%p A328158 a:= n-> add(b(2*n$2, n-i)*(-1)^i*binomial(n, i), i=0..n):
%p A328158 seq(a(n), n=0..18);
%t A328158 b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i - 1], k] Binomial[k + i j - 1, i j], {j, 0, n/i}]]];
%t A328158 a[n_] := Sum[b[2n, 2n, n-i] (-1)^i Binomial[n, i], {i, 0, n}];
%t A328158 a /@ Range[0, 18] (* _Jean-François Alcover_, May 08 2020, after Maple *)
%Y A328158 Cf. A326500.
%K A328158 nonn
%O A328158 0,2
%A A328158 _Alois P. Heinz_, Oct 05 2019