A103258 G.f. = theta_4(0,x^4)/theta_4(0,x).

1, 2, 4, 8, 12, 20, 32, 48, 72, 106, 152, 216, 304, 420, 576, 784, 1056, 1412, 1876, 2472, 3240, 4224, 5472, 7056, 9056, 11566, 14712, 18640, 23520, 29572, 37056, 46272, 57600, 71488, 88456, 109152, 134332, 164884, 201888, 246608, 300528, 365428, 443392, 536856

Offset: 0

Author

Noureddine Chair, Jan 27 2005

Keywords

• nonn

Comments

G.f. for the number of partitions of 2n in which all odd parts occur with multiplicities 2, 4 or 6. The even parts appear at most three times. E.g., a(8)=12 because 8 = 6+2 = 6+1+1 = 4+4 = 4+2+2 = 4+2+1+1 = 4+1+1+1+1 = 3+3+2 = 3+3+1+1 = 2+2+2+1+1 = 2+2+1+1+1+1 = 2+1+1+1+1+1+1.

Also the number of partitions of 2n in which the even parts appear with 2 types c, c* and with multiplicity 1. The odd parts with multiplicity 4. E.g., a(6)=8 because we have 6, 6*, 42, 42*, 4*2, 4*2*, 21111, 2*1111.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
• Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4-5, 9, 14.
• Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

Crossrefs

Cf. A002448.

Programs

Mathematica

nmax = 60; CoefficientList[Series[Product[(1-x^(4*k)) * (1-x^(8*k-4)) * (1+x^k) / (1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

PARI

{a(n)=local(A); if(n<0,0,A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^4+A)^2/eta(x+A)^2/eta(x^8+A), n))} /* Michael Somos, Feb 10 2005 */

Formula

Expansion of eta(q^2)*eta(q^4)^2/(eta(q)^2*eta(q^8)) in powers of q.

Euler transform of period 8 sequence [2, 1, 2, -1, 2, 1, 2, 0, ...]. - Michael Somos, Feb 10 2005

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

From Vaclav Kotesovec, Jan 10 2017: (Start)

a(n) ~ sqrt(3)*Pi * BesselI(1, sqrt(3*n)*Pi/2) / (8*sqrt(n)).

a(n) ~ 3^(1/4) * exp(Pi*sqrt(3*n)/2) / (8*n^(3/4)) * (1 - sqrt(3)/(4*Pi*sqrt(n)) - 5/(32*Pi^2*n)).

(End)