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A241865 Number of compositions of n such that the smallest part has multiplicity five.

%I A241865 #14 Feb 09 2015 09:28:59
%S A241865 1,0,6,6,27,49,125,258,579,1202,2512,5157,10463,20949,41627,81912,
%T A241865 159834,309641,595836,1139211,2165502,4094219,7701857,14420351,
%U A241865 26880988,49902183,92279657,170020844,312173822,571307477,1042310911,1896039086,3439404321,6222483152
%N A241865 Number of compositions of n such that the smallest part has multiplicity five.
%H A241865 Joerg Arndt, Alois P. Heinz and Vaclav Kotesovec, <a href="/A241865/b241865.txt">Table of n, a(n) for n = 5..1500</a> (first 1000 terms from Joerg Arndt and Alois P. Heinz)
%F A241865 a(n) ~ n^5 * ((1+sqrt(5))/2)^(n-11) / (5^3 * 5!). - _Vaclav Kotesovec_, May 02 2014
%p A241865 b:= proc(n, s) option remember; `if`(n=0, 1,
%p A241865       `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p A241865     end:
%p A241865 a:= proc(n) local k; k:= 5;
%p A241865       add((p->add(coeff(p, x, i)*binomial(i+k, k),
%p A241865        i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
%p A241865     end:
%p A241865 seq(a(n), n=5..40);
%t A241865 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 = 5}, 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, 5, 40}] (* _Jean-François Alcover_, Feb 09 2015, after Maple *)
%Y A241865 Column k=5 of A238342.
%K A241865 nonn
%O A241865 5,3
%A A241865 _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014