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%I #13 Dec 15 2020 16:27:29
%S 1,1,5,41,505,8597,191457,5364837,183744421,7521913845,361544182917,
%T 20109571623693,1278810836639233,92032189911692253,
%U 7430335604308535497,667922294225164998677,66407623510409091454229,7260203111052685954056549,868289612454444952122790277
%N Number of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and all colors of an initial interval of the color palette are used.
%H Alois P. Heinz, <a href="/A325888/b325888.txt">Table of n, a(n) for n = 0..296</a>
%p b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
%p binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
%p end:
%p a:= n-> add(add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
%p seq(a(n), n=0..21);
%t b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k + j - 1, j], {j, 1, n}]];
%t a[n_] := Sum[Sum[b[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
%t a /@ Range[0, 21] (* _Jean-François Alcover_, Dec 15 2020, after _Alois P. Heinz_ *)
%Y Row sums of A321296.
%K nonn
%O 0,3
%A _Alois P. Heinz_, Sep 07 2019
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