A104277 Number of partitions of n in which both even and odd squares occur with multiplicity 1. There is no restriction on the parts which are twice odd squares.

1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 8, 10, 10, 11, 11, 13, 13, 14, 14, 14, 16, 16, 18, 18, 20, 20, 22, 23, 23, 25, 25, 28, 28, 30, 30, 33, 35, 35, 38, 39, 43, 43, 46, 46, 49, 51, 51, 55, 56, 60, 61

Offset

0,5

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Formula

G.f.: Product_{k>0} (1+x^((2*k)^2))/(1-x^((2*k-1)^2)).

a(n) ~ ((4 - sqrt(2))*zeta(3/2))^(1/3) * exp(3*Pi^(1/3) * ((4 - sqrt(2))*zeta(3/2))^(2/3) * n^(1/3) / 2^(8/3)) / (2^(17/6) * sqrt(3) * Pi^(1/3) * n^(5/6)). - Vaclav Kotesovec, Mar 11 2026

Example

a(21)=7 because we can write 21 as 18+2+1 = 16+4+1 = 16+2+2+1 = 9+4+2+2+2+2 = 9+2+2+2+2+2+2 = 4+2+2+2+2+2+2+2+2+1 = 2+2+2+2+2+2+2+2+2+2+1.

Maple

series(product((1+x^((2*k)^2))/(1-x^((2*k-1)^2)), k=1..100), x=0, 100);

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^((2*k)^2)) / (1 - x^((2*k-1)^2)), {k, 1, Sqrt[nmax]/2 + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 11 2026 *)

Crossrefs

Cf. A167661.

Sequence in context: A290726 A090663 A111890 * A125893 A005857 A025809

Adjacent sequences: A104274 A104275 A104276 * A104278 A104279 A104280

Keyword

easy,nonn

Author

Noureddine Chair, Mar 01 2005

Extensions

Missing term a(46) added by Jason Yuen, Jan 20 2025

Status

approved