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A241866 Number of compositions of n such that the smallest part has multiplicity six.

%I A241866 #10 Feb 09 2015 09:29:04
%S A241866 1,0,7,7,35,63,176,371,861,1862,4032,8512,17851,36848,75286,152334,
%T A241866 305466,607313,1198443,2348388,4571728,8846314,17021480,32579029,
%U A241866 62048589,117627699,222018034,417326148,781395064,1457684326,2709797693,5020734691,9273107977
%N A241866 Number of compositions of n such that the smallest part has multiplicity six.
%H A241866 Joerg Arndt and Alois P. Heinz, <a href="/A241866/b241866.txt">Table of n, a(n) for n = 6..1000</a>
%F A241866 a(n) ~ n^6 * ((1+sqrt(5))/2)^(n-13) / (5^(7/2) * 6!). - _Vaclav Kotesovec_, May 02 2014
%p A241866 b:= proc(n, s) option remember; `if`(n=0, 1,
%p A241866       `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p A241866     end:
%p A241866 a:= proc(n) local k; k:= 6;
%p A241866       add((p->add(coeff(p, x, i)*binomial(i+k, k),
%p A241866        i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
%p A241866     end:
%p A241866 seq(a(n), n=6..40);
%t A241866 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 = 6}, 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, 6, 40}] (* _Jean-François Alcover_, Feb 09 2015, after Maple *)
%Y A241866 Column k=6 of A238342.
%K A241866 nonn
%O A241866 6,3
%A A241866 _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014