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A325478 Number of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and all colors of an initial interval of the color palette are used. 2

%I #20 Dec 14 2020 08:39:58

%S 1,1,4,29,329,5252,110955,2972769,97922354,3872594811,180459028989,

%T 9759149087646,604841170643957,42508077480226893,3357224252026104140,

%U 295651782273190911233,28834727303442640011901,3095877335697619795977036,363977673792652615285223095

%N Number of colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and all colors of an initial interval of the color palette are used.

%H Alois P. Heinz, <a href="/A325478/b325478.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), j=1..min(k, 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..23);

%t b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[b[n - j, k] Binomial[n - 1, j - 1] Binomial[k, j], {j, 1, Min[k, 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, 23] (* _Jean-François Alcover_, Dec 14 2020, after _Alois P. Heinz_ *)

%Y Row sums of A322670.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Sep 06 2019

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Last modified March 28 08:19 EDT 2024. Contains 371236 sequences. (Running on oeis4.)