A363045 Number of partitions of n whose greatest part is a multiple of 3.
1, 0, 0, 1, 1, 2, 4, 5, 7, 11, 14, 19, 27, 34, 45, 60, 77, 99, 130, 163, 208, 265, 333, 417, 526, 651, 810, 1004, 1237, 1519, 1869, 2278, 2780, 3382, 4101, 4958, 5995, 7210, 8669, 10398, 12444, 14859, 17730, 21086, 25057, 29718, 35186, 41584, 49100, 57842, 68075, 79991
Offset: 0
Keywords
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i)))) end: a:= n-> add(b(n-3*i, min(n-3*i, 3*i)), i=0..n/3): seq(a(n), n=0..60); # Alois P. Heinz, May 14 2023 -
Mathematica
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]]; a[n_] := Sum[b[n - 3*i, Min[n - 3*i, 3*i]], {i, 0, n/3}]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Oct 23 2023, after Alois P. Heinz *) -
PARI
my(N=60, x='x+O('x^N)); Vec(sum(k=0, N, x^(3*k)/prod(j=1, 3*k, 1-x^j)))
Formula
G.f.: Sum_{k>=0} x^(3*k)/Product_{j=1..3*k} (1-x^j).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (12*sqrt(3)*n) * (1 - (1/Pi + Pi/72)*sqrt(3/(2*n))). - Vaclav Kotesovec, May 20 2023