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%I #9 Apr 11 2022 09:12:59
%S 1,1,60,7512,1546042,541742985,267920998650,180675370176420,
%T 160654598650809964,178879511446386682365,243695196628845859469020,
%U 400544315906804782687318938,777083567062772102871149374020,1755895011129198763056241198051342
%N Number of colored compositions of 2n using all colors of an n-set such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order).
%H Alois P. Heinz, <a href="/A327678/b327678.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = A327673(2n,n).
%p b:= proc(n, i, k, p) option remember;
%p `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
%p b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
%p end:
%p a:= n-> add(b(2*n$2, i, 0)*(-1)^(n-i)*binomial(n, i), i=0..n):
%p seq(a(n), n=0..15);
%t b[n_, i_, k_, p_] := b[n, i, k, p] =
%t If[n == 0, p!, If[i < 1, 0, Sum[Binomial[k^i, j]*
%t b[n - i*j, Min[n - i*j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
%t a[n_] := Sum[b[2n, 2n, i, 0]*(-1)^(n-i)*Binomial[n, i], {i, 0, n}];
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 11 2022, after _Alois P. Heinz_ *)
%Y Cf. A327673.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 21 2019
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