A335746 a(n) is the number of partitions of n into distinct parts such that the number of parts divisible by 3 is even.

1, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 32, 38, 45, 52, 61, 71, 83, 96, 111, 128, 148, 170, 195, 224, 256, 293, 334, 380, 432, 491, 557, 630, 713, 805, 908, 1024, 1152, 1295, 1455, 1632, 1829, 2049, 2291, 2560, 2859, 3189, 3554, 3959, 4404, 4896, 5440

Offset

0,6

Links

Table of n, a(n) for n=0..60.

J. Huh and B. Kim, The number of equivalence classes arising from partition involutions, Int. J. Number Theory, 16 (2020), 925-939.

Formula

G.f.: 1/2*(Product_{n>=1} (1+q^n) + Product_{n>=1} (1+q^(3*n-2))*(1+q^(3*n-1))*(1-q^(3*n))).

a(n) = (A000009(n) + A080995(n))/2. - Vaclav Kotesovec, Nov 15 2024

Example

a(10) = 5, the relevant partitions of 10 into distinct parts being [10], [8,2], [7,2,1], [6,3,1], [5,4,1].

Prog

(PARI) seq(n)={my(A=O(x*x^n)); Vec(prod(k=1, n, 1 + x^k + A) + prod(k=1, ceil(n/3), (1+x^(3*k-2)+A)*(1+x^(3*k-1)+A)*(1-x^(3*k)+A)))/2} \\ Andrew Howroyd, Jul 02 2020

Crossrefs

Cf. A000009, A080995.

Sequence in context: A029021 A261770 A096792 * A015741 A015753 A005686

Adjacent sequences: A335743 A335744 A335745 * A335747 A335748 A335749

Keyword

nonn

Author

Jeremy Lovejoy, Jul 02 2020

Status

approved