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%I #10 Feb 09 2015 09:29:16
%S 1,0,9,9,54,99,309,684,1720,3909,9036,20178,44676,97191,209151,444498,
%T 935002,1947729,4021429,8234244,16732173,33758283,67656843,134751630,
%U 266817214,525411981,1029271671,2006453683,3893241810,7521104292,14468931402,27724579185
%N Number of compositions of n such that the smallest part has multiplicity eight.
%H Joerg Arndt and Alois P. Heinz, <a href="/A241868/b241868.txt">Table of n, a(n) for n = 8..1000</a>
%F a(n) ~ n^8 * ((1+sqrt(5))/2)^(n-17) / (5^(9/2) * 8!). - _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:= 8;
%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=8..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 = 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 *)
%Y Column k=8 of A238342.
%K nonn
%O 8,3
%A _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014
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