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A320320 Number of weakly unimodal compositions of n in which the greatest part occurs exactly nine times.

%I A320320 #6 Oct 24 2018 16:28:49
%S A320320 1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,11,13,17,21,28,
%T A320320 35,46,57,73,90,113,138,172,209,258,314,385,467,572,692,843,1020,1238,
%U A320320 1493,1808,2175,2624,3152,3790,4540,5447,6509,7786,9287,11080
%N A320320 Number of weakly unimodal compositions of n in which the greatest part occurs exactly nine times.
%H A320320 Alois P. Heinz, <a href="/A320320/b320320.txt">Table of n, a(n) for n = 0..10000</a>
%H A320320 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unimodality#Unimodal_function">Unimodality, Unimodal function</a>
%F A320320 G.f.: Sum_{n>=0} x^(9*n) / Product_{j=1..n-1} (1-x^j)^2.
%F A320320 a(n) ~ Pi^8 * 8! * exp(2*Pi*sqrt(n/3)) / (2^11 * 3^(19/4) * n^(21/4)). - _Vaclav Kotesovec_, Oct 24 2018
%p A320320 b:= proc(n, i) option remember; `if`(i>n, 0,
%p A320320       `if`(9*i=n, 1, 0)+add(b(n-i*j, i+1)*(j+1), j=0..n/i))
%p A320320     end:
%p A320320 a:= n-> `if`(n=0, 1, b(n, 1)):
%p A320320 seq(a(n), n=0..70);
%Y A320320 Column k=9 of A247255.
%K A320320 nonn
%O A320320 0,20
%A A320320 _Alois P. Heinz_, Oct 10 2018