A327680 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 parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order.

0, 1, 7, 44, 358, 2904, 29112, 296448, 3520568, 43482208, 602603120, 8712724080, 138736978208, 2302036052128, 41417364992160, 776413790063328, 15597709327298944, 325945020056535968, 7238587734613470208, 166897326948551436384, 4061690336695535982048

Offset: 0

Author

Alois P. Heinz, Sep 21 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A309973.

Programs

Maple

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)*
       binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))
    end:
a:= n-> add(add(k*b(n$2, i)*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..22);

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-1, i], j] j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[k b[n, n, i](-1)^(k-i)Binomial[k, i], {i, 0, k}], {k, 0, n}];
a /@ Range[0, 22] (* Jean-François Alcover, Dec 18 2020, after_Alois P. Heinz_ *)

Formula

a(n) = Sum_{k=1..n} k * A309973(n,k).