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

0,3

Links

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

Formula

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

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_ *)

Crossrefs

Cf. A309973.

Sequence in context: A091127 A166775 A221541 * A156374 A171493 A262772

Adjacent sequences: A327677 A327678 A327679 * A327681 A327682 A327683

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2019

Status

approved