A241870 Number of compositions of n such that the smallest part has multiplicity ten.

1, 0, 11, 11, 77, 143, 495, 1133, 3058, 7271, 17777, 41580, 96701, 220187, 495528, 1099626, 2412927, 5236308, 11251449, 23952841, 50556265, 105852923, 219975999, 453933348, 930544912, 1895736986, 3839424644, 7732852963, 15492659226, 30884561378, 61276442019

Offset: 10

Author

Joerg Arndt and Alois P. Heinz, Apr 30 2014

Keywords

• nonn

Comments

Conjecture: Generally, for k > 0 is column k of A238342 asymptotic to n^k * ((1+sqrt(5))/2)^(n-2*k-1) / (5^((k+1)/2) * k!). - Vaclav Kotesovec, May 02 2014

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 10..1000

Crossrefs

Column k=10 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=10..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 = 10}, 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, 10, 40}] (* Jean-François Alcover, Feb 09 2015, after Maple *)

Formula

a(n) ~ n^10 * ((1+sqrt(5))/2)^(n-21) / (5^(11/2) * 10!). - Vaclav Kotesovec, May 02 2014