A326649 Total number of colors in all colored integer partitions of n using all colors of an initial interval of the color palette such that each block of part i with multiplicity j has a pattern of i*j distinct colors in increasing order.

0, 1, 4, 19, 81, 413, 2439, 14655, 86844, 573196, 4224230, 32280154, 249433713, 1925416359, 15732592327, 139542345546, 1304524118159, 12445507282579, 119198874300879, 1137647406084952, 11183828252431175, 116368970786569604, 1278400213028604214

Offset: 0

Author

Alois P. Heinz, Sep 12 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A185283, A326616, A326617.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 0, numtheory[sigma](n)+g(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 b(n-t, min(n-t, i-1), k)*binomial(k, t))(i*j), j=0..n/i)))
    end:
a:= n-> add(k*add(b(n$2, k-i)*(-1)^i*binomial(k, i), i=0..k), k=h(n)..n):
seq(a(n), n=0..25);

Mathematica

g[n_] := g[n] = If[n == 0, 0, DivisorSigma[1, n] + g[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 || kJean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=A185283(n)..n} k * A326616(n,k) = Sum_{k=A185283(n)..n} k * A326617(n,k).