A329384 G.f.: (1 + x) * (1 + x^2) * (1 + x^3) * Product_{k>=1} (1 + x^k).

1, 2, 3, 6, 8, 11, 16, 20, 26, 34, 43, 54, 68, 84, 103, 127, 154, 186, 225, 269, 321, 383, 453, 535, 631, 740, 866, 1012, 1178, 1368, 1587, 1835, 2117, 2440, 2804, 3217, 3687, 4215, 4812, 5487, 6244, 7096, 8055, 9128, 10331, 11681, 13187, 14870, 16752, 18846, 21180

Offset: 0

Author

Ilya Gutkovskiy, Jun 07 2020

Keywords

• nonn

Comments

Number of partitions of n into distinct parts if there are two types of 1's, two types of 2's and two types of 3's.

Crossrefs

Cf. A000009, A000098, A022567, A036469, A052816, A329289.

Programs

Mathematica

nmax = 50; CoefficientList[Series[(1 + x) (1 + x^2) (1 + x^3) Product[(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = (1/n) Sum[Sum[(-1)^(k/d + 1) If[d < 4, 2, 1] d, {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 50}]

Formula

a(n) = A036469(n) + A036469(n-3) - A036469(n-4) - A036469(n-7).

a(n) ~ 2*exp(Pi*sqrt(n/3)) / (3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jun 11 2020