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

%I A241870 #12 Feb 09 2015 09:29:28
%S A241870 1,0,11,11,77,143,495,1133,3058,7271,17777,41580,96701,220187,495528,
%T A241870 1099626,2412927,5236308,11251449,23952841,50556265,105852923,
%U A241870 219975999,453933348,930544912,1895736986,3839424644,7732852963,15492659226,30884561378,61276442019
%N A241870 Number of compositions of n such that the smallest part has multiplicity ten.
%C A241870 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
%H A241870 Joerg Arndt and Alois P. Heinz, <a href="/A241870/b241870.txt">Table of n, a(n) for n = 10..1000</a>
%F A241870 a(n) ~ n^10 * ((1+sqrt(5))/2)^(n-21) / (5^(11/2) * 10!). - _Vaclav Kotesovec_, May 02 2014
%p A241870 b:= proc(n, s) option remember; `if`(n=0, 1,
%p A241870       `if`(n<s, 0, expand(add(b(n-j, s)*x, j=s..n))))
%p A241870     end:
%p A241870 a:= proc(n) local k; k:= 10;
%p A241870       add((p->add(coeff(p, x, i)*binomial(i+k, k),
%p A241870        i=0..degree(p)))(b(n-j*k, j+1)), j=1..n/k)
%p A241870     end:
%p A241870 seq(a(n), n=10..40);
%t A241870 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 *)
%Y A241870 Column k=10 of A238342.
%K A241870 nonn
%O A241870 10,3
%A A241870 _Joerg Arndt_ and _Alois P. Heinz_, Apr 30 2014