A329289 Expansion of g.f. (1 + x) * (1 + x^2) * Product_{k>=1} (1 + x^k).

1, 2, 3, 5, 6, 8, 11, 14, 18, 23, 29, 36, 45, 55, 67, 82, 99, 119, 143, 170, 202, 240, 283, 333, 391, 457, 533, 621, 721, 835, 966, 1114, 1282, 1474, 1690, 1935, 2213, 2525, 2877, 3274, 3719, 4219, 4781, 5409, 6112, 6900, 7778, 8758, 9852, 11068, 12422

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..50.

Formula

a(n) = A036469(n) - A036469(n-4).

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

G.f.: A(x) = 1/2 * Sum_{n >= 0} x^((n-2)*(n-3)/2) / (Product_{k = 1..n} 1 - x^k). - Peter Bala, Feb 03 2025

Maple

series(1/2 * add( x^((n-2)*(n-3)/2) / mul(1 - x^k, k = 1..n), n = 0..12), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Feb 03 2025

Mathematica

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

Crossrefs

Cf. A000009, A000097, A022567, A036469, A052816, A096914, A329384.

Sequence in context: A127312 A081830 A238006 * A385067 A117517 A339732

Adjacent sequences: A329286 A329287 A329288 * A329290 A329291 A329292

Keyword

nonn,easy

Author

Ilya Gutkovskiy, Jun 07 2020

Status

approved