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A327118 Total number of colors in all colored integer partitions of n using all colors of an initial color palette such that a color pattern for part i has i distinct colors in increasing order. 2
0, 1, 5, 24, 129, 752, 4796, 33117, 246336, 1961233, 16626100, 149376533, 1416602126, 14130107135, 147781380186, 1616110614723, 18434515499407, 218849548323400, 2698686223271769, 34504328470389166, 456669361749612835, 6247290917385938422, 88216873775207493056 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..230

Wikipedia, Partition (number theory)

FORMULA

a(n) = Sum_{k=1..n} k * A327117(n,k).

MAPLE

C:= binomial:

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(

      b(n-i*j, min(n-i*j, i-1), k)*C(C(k, i)+j-1, j), j=0..n/i)))

    end:

a:= n-> add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):

seq(a(n), n=0..25);

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k, i]+j-1, j], {j, 0, n/i}]]];

a[n_] := Sum[k Sum[b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];

a /@ Range[0, 25] (* Jean-Fran├žois Alcover, May 06 2020, after Maple *)

CROSSREFS

Cf. A327117.

Sequence in context: A275267 A278679 A000349 * A036919 A020067 A066118

Adjacent sequences:  A327115 A327116 A327117 * A327119 A327120 A327121

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Sep 13 2019

STATUS

approved

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Last modified January 24 21:49 EST 2022. Contains 350565 sequences. (Running on oeis4.)