A241868 Number of compositions of n such that the smallest part has multiplicity eight.

1, 0, 9, 9, 54, 99, 309, 684, 1720, 3909, 9036, 20178, 44676, 97191, 209151, 444498, 935002, 1947729, 4021429, 8234244, 16732173, 33758283, 67656843, 134751630, 266817214, 525411981, 1029271671, 2006453683, 3893241810, 7521104292, 14468931402, 27724579185

Offset: 8

Author

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Keywords

• nonn

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 8..1000

Crossrefs

Column k=8 of A238342.

Programs

Maple

b:= proc(n, s) option remember; `if`(n=0, 1,
      `if`(nadd(coeff(p, x, i)*binomial(i+k, k),
       i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
    end:
seq(a(n), n=8..40);

Mathematica

b[n_, s_] := b[n, s] = If[n == 0, 1, If[n < s, 0, Expand[Sum[b[n - j, s]*x, {j, s, n}]]]]; a[n_] := With[{k = 8}, Sum[Function[{p}, Sum[Coefficient[p, x, i]*Binomial[i + k, k], {i, 0, Exponent[p, x]}]][b[n - j*k, j + 1]], {j, 1, n/k}]]; Table[a[n], {n, 8, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

Formula

a(n) ~ n^8 * ((1+sqrt(5))/2)^(n-17) / (5^(9/2) * 8!). - Vaclav Kotesovec, May 02 2014