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A363045 Number of partitions of n whose greatest part is a multiple of 3.

%I A363045 #29 Oct 23 2023 04:33:48
%S A363045 1,0,0,1,1,2,4,5,7,11,14,19,27,34,45,60,77,99,130,163,208,265,333,417,
%T A363045 526,651,810,1004,1237,1519,1869,2278,2780,3382,4101,4958,5995,7210,
%U A363045 8669,10398,12444,14859,17730,21086,25057,29718,35186,41584,49100,57842,68075,79991
%N A363045 Number of partitions of n whose greatest part is a multiple of 3.
%H A363045 Vaclav Kotesovec, <a href="/A363045/b363045.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Seiichi Manyama)
%F A363045 G.f.: Sum_{k>=0} x^(3*k)/Product_{j=1..3*k} (1-x^j).
%F A363045 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
%p A363045 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A363045      `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
%p A363045     end:
%p A363045 a:= n-> add(b(n-3*i, min(n-3*i, 3*i)), i=0..n/3):
%p A363045 seq(a(n), n=0..60);  # _Alois P. Heinz_, May 14 2023
%t A363045 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]]]];
%t A363045 a[n_] := Sum[b[n - 3*i, Min[n - 3*i, 3*i]], {i, 0, n/3}];
%t A363045 Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Oct 23 2023, after _Alois P. Heinz_ *)
%o A363045 (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)))
%Y A363045 Column 3 of A363048.
%Y A363045 Cf. A038499, A072233.
%K A363045 nonn
%O A363045 0,6
%A A363045 _Seiichi Manyama_, May 14 2023