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A327843 Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.

%I A327843 #9 Dec 18 2020 04:02:02
%S A327843 1,1,7,94,2081,67390,2969647,169299808,12032189630,1036485156029,
%T A327843 105880393642170,12604896326749405,1724189631362670619,
%U A327843 267831346979691504798,46782781937811822181581,9111872329195713764645644,1964607669245374038857479576
%N A327843 Number of colored integer partitions of 2n using all colors of an n-set such that a color pattern for part i has i distinct colors in increasing order.
%H A327843 Alois P. Heinz, <a href="/A327843/b327843.txt">Table of n, a(n) for n = 0..160</a>
%F A327843 a(n) = A327117(2n,n).
%e A327843 a(2) = 7: 2ab2ab, 2ab1a1a, 2ab1a1b, 2ab1b1b 1a1a1a1b, 1a1a1b1b, 1a1b1b1b.
%p A327843 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A327843       add(b(n-i*j, min(n-i*j, i-1), k)*binomial(
%p A327843       binomial(k, i)+j-1, j), j=0..n/i)))
%p A327843     end:
%p A327843 a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
%p A327843 seq(a(n), n=0..17);
%t A327843 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[Binomial[k, i] + j - 1, j], {j, 0, n/i}]]];
%t A327843 a[n_] := Sum[b[2n, 2n, i] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
%t A327843 a /@ Range[0, 17] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327843 Cf. A327117.
%K A327843 nonn
%O A327843 0,3
%A A327843 _Alois P. Heinz_, Sep 27 2019