A112192 Coefficients of replicable function number "48h".

1, 1, 2, 2, 3, 4, 5, 7, 8, 10, 13, 16, 20, 24, 30, 36, 43, 52, 61, 73, 86, 102, 120, 140, 165, 192, 224, 260, 301, 348, 401, 462, 530, 608, 696, 796, 909, 1035, 1178, 1338, 1518, 1720, 1945, 2198, 2480, 2796, 3148, 3540, 3978, 4464, 5006, 5606, 6273, 7012

Offset

0,3

Comments

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for McKay-Thompson series for Monster simple group

Formula

a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 10 2015

Expansion of f(x^2, x^4) / f(-x, -x^5) in powers of x where f() is Ramanujan's general theta function. - Michael Somos, Sep 30 2015

Expansion of q^(1/2) * eta(q^3) * eta(q^4) / (eta(q) * eta(q^12)) in powers of q. - Michael Somos, Sep 30 2015

Euler transform of period 12 sequence [1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, ...]. - Michael Somos, Sep 30 2015

a(n) = number of partitions of n into parts == +-1, +-2, +-5 (mod 12). - Michael Somos, Sep 30 2015

G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 30 2015

a(n) = A112186(2*n) = A112187(2*n). - Michael Somos, Sep 30 2015

Convolution inverse of A262771. - Michael Somos, Sep 30 2015

Example

G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 7*x^7 + 8*x^8 + ...
G.f. = 1/q + q^3 + 2*q^7 + 2*q^11 + 3*q^15 + 4*q^19 + 5*q^23 + 7*q^27 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1-x^(3*k)) * (1-x^(4*k)) / ((1-x^k) * (1-x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 10 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^4] / ( QPochhammer[ x] QPochhammer[ x^12]), {x, 0, n}]; (* Michael Somos, Sep 30 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)), n))}; /* Michael Somos, Sep 30 2015 */

Crossrefs

Cf. A058578, A187091.

Cf. A112186, A112187, A262771.

Sequence in context: A349149 A029012 A095699 * A017841 A304329 A111901

Adjacent sequences: A112189 A112190 A112191 * A112193 A112194 A112195

Keyword

nonn

Author

Michael Somos, Aug 28 2005

Status

approved