%I #10 Feb 09 2015 09:29:22
%S 1,0,10,10,65,120,395,890,2320,5401,12847,29380,66735,148630,327270,
%T 711247,1529020,3252775,6855276,14320645,29672905,61018010,124587120,
%U 252694835,509337682,1020610708,2033777830,4031514561,7951981550,15611183177,30510678865
%N Number of compositions of n such that the smallest part has multiplicity nine.
%H Joerg Arndt and Alois P. Heinz, <a href="/A241869/b241869.txt">Table of n, a(n) for n = 9..1000</a>
%F a(n) ~ n^9 * ((1+sqrt(5))/2)^(n-19) / (5^5 * 9!). - _Vaclav Kotesovec_, May 02 2014
%p b:= proc(n, s) option remember; `if`(n=0, 1,
%p `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p end:
%p a:= proc(n) local k; k:= 9;
%p add((p->add(coeff(p, x, i)*binomial(i+k, k),
%p i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
%p end:
%p seq(a(n), n=9..40);
%t 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=9}, 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, 9, 40}] (* _Jean-François Alcover_, Feb 09 2015, after Maple *)
%Y Column k=9 of A238342.
%K nonn
%O 9,3
%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014