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A320316 Number of weakly unimodal compositions of n in which the greatest part occurs exactly five times.

%I A320316 #10 Apr 22 2021 08:44:49
%S A320316 1,0,0,0,0,1,0,0,0,0,1,2,3,4,5,7,9,13,17,24,32,44,58,79,103,138,180,
%T A320316 237,307,402,517,670,859,1104,1407,1799,2280,2896,3656,4616,5801,7291,
%U A320316 9120,11407,14215,17701,21971,27252,33699,41637,51314,63170,77590,95202
%N A320316 Number of weakly unimodal compositions of n in which the greatest part occurs exactly five times.
%H A320316 Alois P. Heinz, <a href="/A320316/b320316.txt">Table of n, a(n) for n = 0..10000</a>
%H A320316 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unimodality#Unimodal_function">Unimodality, Unimodal function</a>
%F A320316 G.f.: Sum_{n>=0} x^(5*n) / Product_{j=1..n-1} (1-x^j)^2.
%F A320316 a(n) ~ Pi^4 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(13/4)). - _Vaclav Kotesovec_, Oct 24 2018
%p A320316 b:= proc(n, i) option remember; `if`(i>n, 0,
%p A320316       `if`(5*i=n, 1, 0)+add(b(n-i*j, i+1)*(j+1), j=0..n/i))
%p A320316     end:
%p A320316 a:= n-> `if`(n=0, 1, b(n, 1)):
%p A320316 seq(a(n), n=0..70);
%t A320316 b[n_, i_] := b[n, i] = If[i > n, 0, If[5 i == n, 1, 0] +
%t A320316      Sum[b[n - i j, i + 1] (j + 1), {j, 0, n/i}]];
%t A320316 a[n_] := If[n == 0, 1, b[n, 1]];
%t A320316 a /@ Range[0, 70] (* _Jean-François Alcover_, Apr 22 2021, after _Alois P. Heinz_ *)
%Y A320316 Column k=5 of A247255.
%K A320316 nonn
%O A320316 0,12
%A A320316 _Alois P. Heinz_, Oct 10 2018