A104276 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 even squares.

1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 1, 2, 1, 0, 2, 3, 1, 0, 2, 3, 1, 0, 2, 5, 3, 0, 2, 5, 3, 0, 3, 6, 4, 1, 4, 7, 4, 1, 4, 9, 6, 1, 4, 10, 7, 1, 5, 12, 9, 2, 6, 13, 9, 2, 6, 15, 12, 3, 6, 17, 14, 3, 8, 20, 16, 4, 9, 21, 17, 5, 10, 25, 22, 7, 10, 27, 24, 7, 12, 32, 28, 9, 14, 34, 30, 10, 15, 39, 37

Offset

0,10

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..20000

Formula

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

a(n) ~ (7*zeta(3/2))^(2/3) * exp(3*Pi^(1/3) * ((4 - sqrt(2))*zeta(3/2))^(2/3) * n^(1/3)/2^(8/3)) / (4*(4 + sqrt(2))^(2/3) * sqrt(3) * Pi^(7/6) * n^(7/6)). - Vaclav Kotesovec, Mar 11 2026

Example

E.g. a(30) = 3 because we can write 30 as 25+4+1 = 16+9+4+1 = 8+8+9+4+1.

Maple

S := series(product((1+x^((2*k-1)^2))/(1-x^((2*k)^2)), k=1..100), x=0, 100):
seq(coeff(S, x, n), n=0..90);

Mathematica

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

Crossrefs

Cf. A167700.

Sequence in context: A376648 A106262 A025870 * A216191 A268834 A285914

Adjacent sequences: A104273 A104274 A104275 * A104277 A104278 A104279

Keyword

nonn

Author

Noureddine Chair, Mar 01 2005

Status

approved