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A098151 Number of partitions of 2n prime to 3 with all odd parts occurring with even multiplicities. There is no restriction on the even parts. 20
1, 2, 4, 6, 10, 16, 24, 36, 52, 74, 104, 144, 198, 268, 360, 480, 634, 832, 1084, 1404, 1808, 2316, 2952, 3744, 4728, 5946, 7448, 9294, 11556, 14320, 17688, 21780, 26740, 32736, 39968, 48672, 59122, 71644, 86616, 104484, 125768, 151072, 181104, 216684 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - Michael Somos, Apr 15 2012

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

a(n) is also the number of Schur overpartitions of n, i.e., the number of overpartitions of n where adjacent parts differ by at least 3 if the smaller is overlined or divisible by 3 and adjacent parts differ by at least 6 if the smaller is overlined and divisible by 3. - Jeremy Lovejoy, Mar 23 2015

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016

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

Jeremy Lovejoy, A theorem on seven-colored overpartitions and its applications, Int. J. Number Theory. 1 (2005) 215-224

Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.

Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Apr 15 2012

Expansion of f(q, q^2) / f(-q, -q^2) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 15 2012

Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.

G.f. = (Sum((-1)^n*q^(3*n^2),n=-oo..oo)) /(Sum((-1)^n*q^(n^2),n=-oo..oo)). - N. J. A. Sloane, Aug 09 2016

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1 + u^2) * (u^2 + v^4) - 4 * u^2*v^4. - Michael Somos, Apr 15 2012

G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u^3 - v + 3 * u*v^2 - 3 * u^2*v^3. - Michael Somos, Dec 04 2004

Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - Vladeta Jovovic, Sep 24 2004

Taylor series of product_{k=1..inf}(1+x^k+x^(2*k))/(1-x^k+x^(2*k))= product_{k=1..inf}(1+x^k)(1-x^(3k))/((1-x^k)(1+x^(3k)))=Theta_4(0, x^3)/theta_4(0, x)

a(n) ~ Pi * BesselI(1, Pi*sqrt(2*n/3)) / (3*sqrt(2*n)) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/4) * 3^(3/4) * n^(3/4)) * (1 - 3*sqrt(3)/(8*Pi*sqrt(2*n)) - 45/(256*Pi^2*n)). - Vaclav Kotesovec, Aug 31 2015, extended Jan 09 2017

Convolution of A000726 and A003105. - R. J. Mathar, Nov 17 2017

EXAMPLE

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

G.f. = 1 + 2*q + 4*q^2 + 6*q^3 + 10*q^4 + 16*q^5 + 24*q^6 + 36*q^7 + 52*q^8 + ...

MAPLE

series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)), k=1..150), x=0, 100);

# alternative program using expansion of f(q, q^2) / f(-q, -q^2):

with(gfun): series( add(x^(n*(3*n-1)/2), n = -8..8)/add((-1)^n*x^(n*(3*n-1)/2), n = -8..8), x, 100): seriestolist(%); # Peter Bala, Feb 05 2021

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* Michael Somos, Oct 23 2013 *)

nmax = 50; CoefficientList[Series[Product[(1+x^(3*k-1)) * (1+x^(3*k-2)) / ((1-x^(3*k-1)) * (1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 / (eta(x + A)^2 * eta(x^6 + A)), n))} /* Michael Somos, Dec 04 2004 */

CROSSREFS

Cf. A015128, A001318, A080054, A080995, A010815, A103260.

Sequence in context: A261204 A293422 A132002 * A137414 A211971 A305498

Adjacent sequences:  A098148 A098149 A098150 * A098152 A098153 A098154

KEYWORD

nonn,easy

AUTHOR

Noureddine Chair, Aug 29 2004

STATUS

approved

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Last modified July 27 18:27 EDT 2021. Contains 346308 sequences. (Running on oeis4.)