STATUS
proposed
approved
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Revision History for A325930
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Total number of colors used in all colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and the colors span an initial interval of the color palette.
(history; published version)
(history; published version)
STATUS
editing
proposed
MATHEMATICA
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]}]];
a[n_] := Sum[k Sum[b[n, k-i] (-1)^i Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 18] (* Jean-François Alcover, Dec 15 2020, after Alois P. Heinz *)
STATUS
approved
editing
STATUS
editing
approved
LINKS
Alois P. Heinz, <a href="/A325930/b325930.txt">Table of n, a(n) for n = 0..296</a>
MAPLE
b:= proc(n, k) option remember; `if`(n=0, 1, add(b(n-j, k)*
binomial(n-1, j-1)*binomial(k, j), j=1..min(k, n)))
end:
a:= n-> add(k*add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..18);
NAME
a
Total number of colors used in all colored set partitions of [n] where colors of the elements of subsets are distinct and in increasing order and the colors span an initial interval of the color palette.
DATA
0, 1, 7, 73, 1075, 21066, 527122, 16313963, 609352653, 26938878757, 1387465470527, 82169954359252, 5534425340505464, 419977314311140561, 35617039966665620743, 3352008343756176938273, 347915661537105210844323, 39607489635223003610928042
OFFSET
0,13
NAME
allocated for Alois P. Heinz
a
DATA
73, 1075, 21066, 527122
OFFSET
0,1
FORMULA
a(n) = Sum_{k=1..n} k * A322670(n,k).
CROSSREFS
Cf. A322670.
KEYWORD
allocated
nonn
AUTHOR
Alois P. Heinz, Sep 08 2019
STATUS
approved
editing
NAME
allocated for Alois P. Heinz
KEYWORD
recycled
allocated
STATUS
editing
approved