A097242 - OEIS

%I #46 Jun 21 2020 03:51:47

%S 1,0,1,1,1,1,2,1,2,3,3,3,5,4,6,7,7,8,11,10,13,15,16,18,23,22,27,31,33,

%T 37,45,45,53,60,64,71,84,86,99,111,119,131,151,157,178,198,212,233,

%U 264,277,310,342,367,401,449,474,525,576,618,673,746,790,869,949,1017,1104

%N Expansion of q-series 1 / (q^2, q^3, q^9, q^10; q^12)_infinity.

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

%C Number of partitions of n into odd parts in which every part occurs at least twice. Example: a(9)=3 because we have [3,3,3], [3,3,1,1,1] and [1,1,1,1,1,1,1,1,1]. - _Vladeta Jovovic_, Jan 16 2005

%C Also equal to the number of partitions of n into distinct parts not congruent to 1 or 5 modulo 6.   Example: a(9) = 3, the relevant partitions being [9], [6,3], and [4,3,2]. - _Jeremy Lovejoy_, Jun 21 2020

%C From _Joerg Arndt_, Jun 21 2020: (Start)

%C a(n) is the number of partitions with parts == { 2, 3, 9, 10 } (mod 12).

%C a(n) is the number of overpartitions with non-overlined parts == 2 (mod 4) and overlined parts == 3 (mod 6); same as the number of partitions with parts == 2 (mod 4) and distinct parts == 3 (mod 6).  (End)

%H Vaclav Kotesovec, <a href="/A097242/b097242.txt">Table of n, a(n) for n = 0..5000</a>

%H K. Alladi, G.E. Andrews, and B. Gordon, <a href="https://doi.org/10.1006/jabr.1995.1144">Refinements and generalizations of Capparelli's conjecture on partitions</a>, Journal of Algebra, 174 (1995), 636-658.

%H J. Dousse and J. Lovejoy, <a href="https://doi.org/10.1112/blms.12218">Generalizations of Capparelli's identity</a>, Bulletin of the London Mathematical Society, 51 (2019), 193-206.

%H J. Lovejoy, <a href="https://doi.org/10.1007/978-3-319-68376-8_26">Asymmetric generalizations of Schur's theorem</a>, in: Andrews G., Garvan F. (eds) Analytic Number Theory, Modular Forms and q-Hypergeometric Series. ALLADI60 2016. Springer Proceedings in Mathematics & Statistics, vol 221. Springer, Cham.

%H A. V. Sills, <a href="http://dx.doi.org/10.1016/j.aam.2003.09.005">On series expansion of Capparelli's infinite product</a>, Adv. in Appl. Math., 33 (2004), pp. 397-408.

%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 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 G.f.: Product_{k>0} (1 + x^(6*k - 3)) / (1 - x^(4*k - 2)).

%F G.f.: 1 / (Product_{k>=0} (1 - x^(12*k + 2)) * (1 - x^(12*k + 3)) * (1 - x^(12*k + 9)) * (1 - x^(12*k + 10))).

%F Expansion of chi(x^3) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Oct 17 2006

%F Expansion of q^(1/24) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.

%F Euler transform of period 12 sequence [0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, ...].

%F a(n) ~ Pi*BesselI(1, sqrt(24*n-1)*Pi/(6*sqrt(3))) / sqrt(3*(24*n-1)/2) ~ exp(Pi*sqrt(2*n)/3) / (2^(7/4) * sqrt(3) * n^(3/4)) * (1 - (9/(8*Pi) + Pi/72)/sqrt(2*n) + (5/128 - 135/(256*Pi^2) + Pi^2/20736)/n). - _Vaclav Kotesovec_, Aug 31 2015, extended Jan 09 2017

%e G.f. = 1 + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + 3*x^9 + 3*x^10 + 3*x^11 + ...

%e G.f. = 1/q + q^47 + q^71 + q^95 + q^119 + 2*q^143 + q^167 + 2*q^191 + 3*q^215 + ...

%p g:=product(1+x^(4*j-2)/(1-x^(2*j-1)),j=1..20): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..65); # _Emeric Deutsch_, Feb 23 2006

%t nmax = 100; CoefficientList[Series[Product[1/((1 - x^(12*k + 2)) * (1 - x^(12*k + 3)) * (1 - x^(12*k + 9)) * (1 - x^(12*k + 10))), {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6], {x, 0, n}]; (* _Michael Somos_, Jan 09 2017 *)

%o (PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - x^k * [0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0][(k-1)%12 + 1], 1 + x * O(x^n)), n))};

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))}; /* _Michael Somos_, Oct 17 2006 */

%Y Cf. A053993.

%K nonn

%O 0,7

%A _Michael Somos_, Aug 02 2004