A327676 Total number of colors in all colored compositions of n using all colors of an initial interval of the color palette such that all parts have different color patterns and the patterns for parts i are sorted and have i colors (in arbitrary order).

0, 1, 7, 96, 1120, 17133, 318550, 6174409, 134222746, 3227634290, 86758393637, 2455321082763, 75378081090618, 2465367832093301, 85698931156186485, 3188667759545387936, 124641366048950285559, 5131034234304472735967, 222142129822095164108159

Offset

0,3

Links

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

Formula

a(n) = Sum_{k=1..n} k * A327673(n,k).

Maple

b:= proc(n, i, k, p) option remember;
     `if`(n=0, p!, `if`(i<1, 0, add(binomial(k^i, j)*
      b(n-i*j, min(n-i*j, i-1), k, p+j)/j!, j=0..n/i)))
    end:
a:= n-> add(add(k*b(n$2, i, 0)*(-1)^(k-i)*
        binomial(k, i), i=0..k), k=0..n):
seq(a(n), n=0..23);

Mathematica

b[n_, i_, k_, p_] := b[n, i, k, p] =
     If[n == 0, p!, If[i < 1, 0, Sum[Binomial[k^i, j]*
b[n - i*j, Min[n - i*j, i - 1], k, p + j]/j!, {j, 0, n/i}]]];
a[n_] := Sum[Sum[k*b[n, n, i, 0]*(-1)^(k-i)*
     Binomial[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

Crossrefs

Cf. A327673.

Sequence in context: A054932 A217240 A222834 * A338075 A177879 A090687

Adjacent sequences: A327673 A327674 A327675 * A327677 A327678 A327679

Keyword

nonn

Author

Alois P. Heinz, Sep 21 2019

Status

approved