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.

0, 1, 5, 24, 129, 752, 4796, 33117, 246336, 1961233, 16626100, 149376533, 1416602126, 14130107135, 147781380186, 1616110614723, 18434515499407, 218849548323400, 2698686223271769, 34504328470389166, 456669361749612835, 6247290917385938422, 88216873775207493056

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-François Alcover, May 06 2020, after Maple *)

Crossrefs

Cf. A327117.

Sequence in context: A275267 A278679 A000349 * A353735 A036919 A020067

Adjacent sequences: A327115 A327116 A327117 * A327119 A327120 A327121

Keyword

nonn

Author

Alois P. Heinz, Sep 13 2019

Status

approved