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A320316
Number of weakly unimodal compositions of n in which the greatest part occurs exactly five times.
2
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, 237, 307, 402, 517, 670, 859, 1104, 1407, 1799, 2280, 2896, 3656, 4616, 5801, 7291, 9120, 11407, 14215, 17701, 21971, 27252, 33699, 41637, 51314, 63170, 77590, 95202
OFFSET
0,12
FORMULA
G.f.: Sum_{n>=0} x^(5*n) / Product_{j=1..n-1} (1-x^j)^2.
a(n) ~ Pi^4 * exp(2*Pi*sqrt(n/3)) / (16 * 3^(7/4) * n^(13/4)). - Vaclav Kotesovec, Oct 24 2018
MAPLE
b:= proc(n, i) option remember; `if`(i>n, 0,
`if`(5*i=n, 1, 0)+add(b(n-i*j, i+1)*(j+1), j=0..n/i))
end:
a:= n-> `if`(n=0, 1, b(n, 1)):
seq(a(n), n=0..70);
MATHEMATICA
b[n_, i_] := b[n, i] = If[i > n, 0, If[5 i == n, 1, 0] +
Sum[b[n - i j, i + 1] (j + 1), {j, 0, n/i}]];
a[n_] := If[n == 0, 1, b[n, 1]];
a /@ Range[0, 70] (* Jean-François Alcover, Apr 22 2021, after Alois P. Heinz *)
CROSSREFS
Column k=5 of A247255.
Sequence in context: A055167 A064628 A188674 * A236166 A017834 A286225
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 10 2018
STATUS
approved