A335745 a(n) is the number of partitions of n without repeated odd parts such that the total number of parts congruent to 0,3, or 5 modulo 8 is even.

1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 11, 15, 18, 23, 28, 37, 44, 55, 66, 83, 99, 121, 145, 179, 212, 255, 303, 366, 431, 514, 606, 723, 847, 999, 1169, 1380, 1607, 1881, 2189, 2561, 2967, 3449, 3990, 4633, 5342, 6172, 7105, 8201, 9412, 10816, 12394, 14225, 16257, 18596, 21220

Offset

0,5

Links

Table of n, a(n) for n=0..55.

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^(2*n-1))/(1-q^(2*n)) + Product_{n>=1} (1+q^(8*n-7))*(1-q^(8*n-5))*(1-q^(8*n-3))*(1+q^(8*n-1))/((1-q^(8*n-6))*(1-q^(8*n-4))*(1-q^(8*n-2))*(1+q^(8*n)))).

Example

a(8) = 6, the relevant partitions of 8 without repeated odd parts being [7,1], [6,2], [5,3], [4,4], [4,2,2], [2,2,2,2].

Crossrefs

Cf. A006950.

Sequence in context: A077117 A237976 A035365 * A379304 A119604 A036806

Adjacent sequences: A335742 A335743 A335744 * A335746 A335747 A335748

Keyword

nonn

Author

Jeremy Lovejoy, Jul 02 2020

Status

approved