A103260 Number of partitions of 2n prime to 3 with all odd parts occurring with multiplicity 2. The even parts occur with multiplicity 1.

1, 2, 2, 2, 2, 4, 6, 8, 10, 10, 12, 16, 22, 28, 32, 36, 42, 52, 66, 80, 92, 104, 120, 144, 174, 206, 236, 266, 304, 356, 420, 488, 554, 624, 708, 816, 946, 1084, 1224, 1372, 1548, 1764, 2016, 2288, 2568, 2868, 3216, 3632, 4110

Offset

0,2

Comments

Convolution of A098884 and A003105. [corrected by Vaclav Kotesovec, Feb 07 2021]

Links

Table of n, a(n) for n=0..48.

Noureddine Chair, Partition Identities From Partial Supersymmetry, arXiv:hep-th/0409011v1, 2004.

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

G.f.: (Theta_4(0, x^2)*theta_4(0, x^3))/(theta_4(0, x)*theta_4(0, x^(6))) = Product_{k>0}((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))).

Euler transform of period 12 sequence [2, -1, 0, 0, 2, 0, 2, 0, 0, -1, 2, 0, ...]. - Vladeta Jovovic, Feb 17 2005

a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 01 2015

G.f.: f(x,x^5)/f(-x,-x^5) = ( Sum_{n = -oo..oo} x^(n*(3*n-2)) )/( Sum_{n = -oo..oo} (-1)^n*x^(n*(3*n-2)) ), where f(a,b) = Sum_{n = -oo..oo} a^(n*(n+1)/2)*b^(n*(n-1)/2) is Ramanujan's 2-variable theta function. Cf. A080054 and A098151. - Peter Bala, Feb 05 2021

Example

E.g. a(7)=8 because 14=10+4=10+2+1+1=8+4+2=8+4+1+1=7+7=5+5+4=5+5+2+1+1.

Maple

series(product(((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))), k=1..100), x=0, 100);
# alternative program:
with(gfun): series( add(x^(n*(3*n-2)), n = -6..6)/add((-1)^n*x^(n*(3*n-2)), n = -6..6), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

Mathematica

nmax = 50; CoefficientList[Series[Product[((1+x^(6*k-1))*(1+x^(6*k-5)))/((1-x^(6*k-1))*(1-x^(6*k-5))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)

Crossrefs

Cf. A003105, A098884, A080054, A001082, A098151.

Sequence in context: A097198 A126663 A032600 * A060824 A334125 A364811

Adjacent sequences: A103257 A103258 A103259 * A103261 A103262 A103263

Keyword

nonn,easy

Author

Noureddine Chair, Feb 15 2005

Extensions

Example corrected by Vaclav Kotesovec, Sep 01 2015

Status

approved