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A327681 Number of colored integer partitions of 2n using all colors of an n-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

%I A327681 #13 Dec 18 2020 04:01:42
%S A327681 1,1,21,619,32621,2619031,298688151,45747815408,9130881915237,
%T A327681 2302153903685914,716914926484850891,270654298469985496639,
%U A327681 121905995767297357401683,64616493201145984241278851,39838866068219563302546530228,28277347692301453998991014108124
%N A327681 Number of colored integer partitions of 2n using all colors of an n-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.
%H A327681 Vaclav Kotesovec, <a href="/A327681/b327681.txt">Table of n, a(n) for n = 0..200</a> (terms 0..100 from Alois P. Heinz)
%F A327681 a(n) = A309973(2n,n).
%p A327681 b:= proc(n, i, k) option remember; `if`(n=0, 1,
%p A327681       `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*
%p A327681        binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
%p A327681     end:
%p A327681 a:= n-> add(b(2*n$2, i)*(-1)^(n-i)*binomial(n, i), i=0..n):
%p A327681 seq(a(n), n=0..17);
%t A327681 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 - 1, i], j]*j!, {j, 0, n/i}]]];
%t A327681 a[n_] := Sum[b[2n, 2n, i] (-1)^(n-i) Binomial[n, i], {i, 0, n}];
%t A327681 a /@ Range[0, 17] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ *)
%Y A327681 Cf. A309973.
%K A327681 nonn
%O A327681 0,3
%A A327681 _Alois P. Heinz_, Sep 21 2019