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%I #55 Dec 18 2023 10:10:01
%S 1,2,4,6,10,16,24,36,52,74,104,144,198,268,360,480,634,832,1084,1404,
%T 1808,2316,2952,3744,4728,5946,7448,9294,11556,14320,17688,21780,
%U 26740,32736,39968,48672,59122,71644,86616,104484,125768,151072,181104,216684
%N Number of partitions of 2n prime to 3 with all odd parts occurring with even multiplicities. There is no restriction on the even parts.
%C There are no partitions of 2n+1 in which all odd parts occur with even multiplicity. - _Michael Somos_, Apr 15 2012
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C 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
%H Vaclav Kotesovec, <a href="/A098151/b098151.txt">Table of n, a(n) for n = 0..2000</a>
%H George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016.
%H Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011, 2004.
%H Jeremy Lovejoy, <a href="http://lovejoy.perso.math.cnrs.fr/overschur3.pdf">A theorem on seven-colored overpartitions and its applications</a>, Int. J. Number Theory. 1 (2005) 215-224
%H Andrew Sills, <a href="https://works.bepress.com/andrew_sills/40/">Rademacher-Type Formulas for Restricted Partition and Overpartition Functions</a>, Ramanujan Journal, 23 (1-3): 253-264, 2010.
%H Andrew Sills, <a href="http://home.dimacs.rutgers.edu/~asills/EMDC/SillsEMDC-Rev.pdf">Towards an Automation of the Circle Method</a>, Gems in Experimental Mathematics in Contemporary Mathematics, 2010, formula S24.
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of phi(-q^3) / phi(-q) in powers of q where phi() is a Ramanujan theta function. - _Michael Somos_, Apr 15 2012
%F 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
%F Expansion of eta(q^2) * eta(q^3)^2 / (eta(q)^2 * eta(q^6)) in powers of q.
%F 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
%F 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
%F 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
%F Euler transform of period 6 sequence [2, 1, 0, 1, 2, 0, ...]. - _Vladeta Jovovic_, Sep 24 2004
%F 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)
%F 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
%F Convolution of A000726 and A003105. - _R. J. Mathar_, Nov 17 2017
%e 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=...
%e 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 + ...
%p series(product((1+x^k+x^(2*k))/(1-x^k+x^(2*k)),k=1..150),x=0,100);
%p # alternative program using expansion of f(q, q^2) / f(-q, -q^2):
%p 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
%t a[ n_] := SeriesCoefficient[ QPochhammer[ q^2] QPochhammer[ q^3]^2 / (QPochhammer[ q]^2 QPochhammer[ q^6]), {q, 0, n}] (* _Michael Somos_, Oct 23 2013 *)
%t 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 *)
%o (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 */
%Y Cf. A015128, A001318, A080054, A080995, A010815, A103260.
%K nonn,easy
%O 0,2
%A _Noureddine Chair_, Aug 29 2004
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