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%I #15 Apr 11 2022 08:10:14
%S 1,1,39,2272,284319,56455146,16786728000,6935657012558,
%T 3810209706509775,2684955985258788274,2361563245536690165774,
%U 2535933313556764621139740,3265213763332455703665035736,4965602758384602312429712415116,8805913731971382862369182854094726
%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 have i colors in (weakly) increasing order.
%H Vaclav Kotesovec, <a href="/A327589/b327589.txt">Table of n, a(n) for n = 0..190</a> (terms 0..120 from Alois P. Heinz)
%F a(n) = A327245(2n,n).
%F a(n) ~ c * d^n * n^(2*n), where d = 1.31520176578651896001... and c = 1.569966657460754514... - _Vaclav Kotesovec_, Sep 19 2019
%p C:= binomial:
%p b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
%p b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, i), j), j=0..n/i)))
%p end:
%p a:= n-> add(b(2*n$2, i, 0)*(-1)^(n-i)*C(n, i), i=0..n):
%p seq(a(n), n=0..15);
%t c = Binomial;
%t b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
%t b[n-i*j, Min[n-i*j, i-1], k, p+j]*c[c[k+i-1, i], j], {j, 0, n/i}]]];
%t a[n_] := Sum[b[2n, 2n, i, 0]*(-1)^(n-i)*c[n, i], {i, 0, n}];
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Apr 11 2022, after _Alois P. Heinz_ *)
%Y Cf. A327245.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 17 2019
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