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

0,3

Links

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

Formula

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

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 *)

Crossrefs

Cf. A327245.

Sequence in context: A180776 A353099 A024089 * A287481 A289212 A060005

Adjacent sequences: A327585 A327586 A327587 * A327589 A327590 A327591

Keyword

nonn

Author

Alois P. Heinz, Sep 17 2019

Status

approved