A327115 Total number of colors used 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 distinct colors in increasing order.

0, 1, 4, 19, 98, 570, 3642, 25292, 189454, 1519648, 12978141, 117437020, 1121299471, 11256640012, 118443403699, 1302670531063, 14938986954323, 178248001223476, 2208487163394749, 28363722744050886, 376991516806826090, 5178009641895235269, 73396161423153313320

Offset: 0

Author

Alois P. Heinz, Sep 13 2019

Keywords

• nonn

Examples

			a(2) = 4: 2ab, 1a1b.  Both colors (a and b) are used twice: 2 + 2 = 4.
		

Links

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

Crossrefs

Cf. A326914, A326962.

Programs

Maple

C:= binomial:
g:= proc(n) option remember; n*2^(n-1) end:
h:= proc(n) option remember; local k; for k from
      `if`(n=0, 0, h(n-1)) do if g(k)>=n then return k fi od
    end:
b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k add(k*add(b(n$2, i)*(-1)^(k-i)*C(k, i), i=0..k), k=h(n)..n):
seq(a(n), n=0..23);

Mathematica

c = Binomial;
g[n_] := g[n] = n 2^(n - 1);
h[n_] := h[n] = Module[{k}, For[k = If[n == 0, 0, h[n - 1]] , True, k++, If [g[k] >= n ,  Return[k]]]];
b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1 || k < h[n], 0, Sum[b[n - i j, Min[n - i j, i - 1], k] c[c[k, 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, h[n], n}];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} A326914(n,k) = Sum_{k=1..n} A326962(n,k).