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A327556 Number of colored integer partitions of 2n using all colors of an n-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

%I A327556 #9 Dec 16 2020 16:04:18
%S A327556 1,1,15,319,10305,456540,26189661,1870454452,161632399892,
%T A327556 16535827882568,1968749174314009,269023182822761584,
%U A327556 41709476698204311667,7266527579101535573799,1410853257166617346437587,303111227353456160724127886,71611509245127165374518157052
%N A327556 Number of colored integer partitions of 2n using all colors of an n-set such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.
%H A327556 Alois P. Heinz, <a href="/A327556/b327556.txt">Table of n, a(n) for n = 0..200</a>
%F A327556 a(n) = A327116(2n,n).
%p A327556 C:= binomial:
%p A327556 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
%p A327556       b(n-i*j, min(n-i*j, i-1), k)*C(C(k+i-1, i), j), j=0..n/i)))
%p A327556     end:
%p A327556 a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*C(n, i), i=0..n):
%p A327556 seq(a(n), n=0..17);
%t A327556 c = Binomial;
%t A327556 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] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
%t A327556 a[n_] := Sum[b[2n, 2n, i] (-1)^(n - i) c[n, i], {i, 0, n}];
%t A327556 a /@ Range[0, 17] (* _Jean-François Alcover_, Dec 16 2020, after _Alois P. Heinz_ *)
%Y A327556 Cf. A327116.
%K A327556 nonn
%O A327556 0,3
%A A327556 _Alois P. Heinz_, Sep 16 2019