A327588 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 have i colors in (weakly) increasing order.

0, 1, 7, 62, 642, 7784, 108824, 1725072, 30605384, 601213744, 12958778704, 304145108160, 7722286425312, 210920029636224, 6166996162239840, 192199468584942816, 6360760834966301120, 222782888877269937664, 8233066075880951824000, 320162458265691237967360

Offset: 0

Author

Alois P. Heinz, Sep 17 2019

Keywords

• nonn

Links

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

Crossrefs

Cf. A327245.

Programs

Maple

C:= binomial:
b:= proc(n, i, k, p) option remember; `if`(n=0, p!, `if`(i<1, 0, add(
      b(n-i*j, min(n-i*j, i-1), k, p+j)*C(C(k+i-1, i), j), j=0..n/i)))
    end:
a:= n-> add(k*add(b(n$2, i, 0)*(-1)^(k-i)*C(k, i), i=0..k), k=0..n):
seq(a(n), n=0..21);

Mathematica

c = Binomial;
b[n_, i_, k_, p_] := b[n, i, k, p] = If[n == 0, p!, If[i < 1, 0, Sum[
     b[n - i*j, Min[n-i*j, i-1], k, p+j]*c[c[k+i-1, i], j], {j, 0, n/i}]]];
a[n_] := Sum[k*Sum[b[n, n, i, 0]*(-1)^(k-i)*c[k, i], {i, 0, k}], {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Apr 11 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=1..n} k * A327245(n,k).