A327557 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and a pattern for part i has i colors in (weakly) increasing order.

0, 1, 5, 29, 173, 1129, 7933, 59757, 480389, 4102233, 37059485, 352891285, 3530465753, 37001007337, 405191214949, 4625525704837, 54929552638957, 677283511701937, 8655757492783861, 114479050583748677, 1564613481125976373, 22068492671782019793

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..300

Formula

a(n) = Sum_{k=1..n} k * A327116(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-1, i), 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..23);

Mathematica

c = Binomial;
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] c[c[k + i - 1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[k Sum[b[n, n, i] (-1)^(k - i) c[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 16 2020, after Alois P. Heinz *)

Crossrefs

Cf. A327116.

Sequence in context: A088349 A272802 A083066 * A163611 A160906 A163073

Adjacent sequences: A327554 A327555 A327556 * A327558 A327559 A327560

Keyword

nonn

Author

Alois P. Heinz, Sep 16 2019

Status

approved