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A097242
Expansion of q-series 1 / (q^2, q^3, q^9, q^10; q^12)_infinity.
7
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, 37, 45, 45, 53, 60, 64, 71, 84, 86, 99, 111, 119, 131, 151, 157, 178, 198, 212, 233, 264, 277, 310, 342, 367, 401, 449, 474, 525, 576, 618, 673, 746, 790, 869, 949, 1017, 1104
OFFSET
0,7
COMMENTS
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
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
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
From Joerg Arndt, Jun 21 2020: (Start)
a(n) is the number of partitions with parts == { 2, 3, 9, 10 } (mod 12).
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)
LINKS
K. Alladi, G.E. Andrews, and B. Gordon, Refinements and generalizations of Capparelli's conjecture on partitions, Journal of Algebra, 174 (1995), 636-658.
J. Dousse and J. Lovejoy, Generalizations of Capparelli's identity, Bulletin of the London Mathematical Society, 51 (2019), 193-206.
J. Lovejoy, Asymmetric generalizations of Schur's theorem, 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.
A. V. Sills, On series expansion of Capparelli's infinite product, Adv. in Appl. Math., 33 (2004), pp. 397-408.
Andrew Sills, Rademacher-Type Formulas for Restricted Partition and Overpartition Functions, Ramanujan Journal, 23 (1-3): 253-264, 2010.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
G.f.: Product_{k>0} (1 + x^(6*k - 3)) / (1 - x^(4*k - 2)).
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))).
Expansion of chi(x^3) / chi(-x^2) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Oct 17 2006
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.
Euler transform of period 12 sequence [0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, ...].
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
Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (1/4) * exp(-Pi/24) * 2^(7/8) * 3^(1/12) * Gamma(2/3)^(2/3) * Gamma(3/4)^(2/3) * (sqrt(2) * (1+3^(1/2)))^(2/3) / Gamma(11/12)^(2/3) / Pi^(1/3) = A388488. - Simon Plouffe, Sep 17 2025
EXAMPLE
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 + ...
G.f. = 1/q + q^47 + q^71 + q^95 + q^119 + 2*q^143 + q^167 + 2*q^191 + 3*q^215 + ...
MAPLE
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
MATHEMATICA
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 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^2] QPochhammer[ -x^3, x^6], {x, 0, n}]; (* Michael Somos, Jan 09 2017 *)
PROG
(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))};
(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 */
CROSSREFS
Cf. A053993.
Sequence in context: A035431 A008681 A362559 * A183202 A306878 A161308
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 02 2004
STATUS
approved