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

0,3

Links

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

Formula

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

Example

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

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<h(n),
      0, add(b(n-i*j, min(n-i*j, i-1), k)*C(C(k, 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=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 *)

Crossrefs

Cf. A326914, A326962.

Sequence in context: A006194 A047099 A211855 * A370024 A306511 A177249

Adjacent sequences: A327112 A327113 A327114 * A327116 A327117 A327118

Keyword

nonn

Author

Alois P. Heinz, Sep 13 2019

Status

approved