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A320315 Number of weakly unimodal compositions of n in which the greatest part occurs exactly four times.

%I A320315 #14 Apr 22 2021 08:42:39
%S A320315 1,0,0,0,1,0,0,0,1,2,3,4,6,8,12,16,24,32,46,62,87,116,160,212,288,380,
%T A320315 508,666,883,1150,1508,1954,2542,3274,4229,5416,6949,8856,11292,14320,
%U A320315 18162,22922,28921,36344,45641,57112,71407,88998,110816,137600,170665
%N A320315 Number of weakly unimodal compositions of n in which the greatest part occurs exactly four times.
%H A320315 Alois P. Heinz, <a href="/A320315/b320315.txt">Table of n, a(n) for n = 0..10000</a>
%H A320315 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unimodality#Unimodal_function">Unimodality, Unimodal function</a>
%F A320315 G.f.: Sum_{n>=0} x^(4*n) / Product_{j=1..n-1} (1-x^j)^2.
%F A320315 a(n) ~ Pi^3 * exp(2*Pi*sqrt(n/3)) / (2^5 * 3^(5/4) * n^(11/4)). - _Vaclav Kotesovec_, Oct 24 2018
%p A320315 b:= proc(n, i) option remember; `if`(i>n, 0,
%p A320315       `if`(4*i=n, 1, 0)+add(b(n-i*j, i+1)*(j+1), j=0..n/i))
%p A320315     end:
%p A320315 a:= n-> `if`(n=0, 1, b(n, 1)):
%p A320315 seq(a(n), n=0..70);
%t A320315 b[n_, i_] := b[n, i] = If[i > n, 0, If[4 i == n, 1, 0] +
%t A320315      Sum[b[n - i j, i + 1] (j + 1), {j, 0, n/i}]];
%t A320315 a[n_] :=  If[n == 0, 1, b[n, 1]];
%t A320315 a /@ Range[0, 70] (* _Jean-François Alcover_, Apr 22 2021, after _Alois P. Heinz_ *)
%Y A320315 Column k=4 of A247255.
%K A320315 nonn
%O A320315 0,10
%A A320315 _Alois P. Heinz_, Oct 10 2018