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A112192 Coefficients of replicable function number "48h". 3
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 (list; graph; refs; listen; history; text; internal format)
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).

M. 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: A211858 A029012 A095699 * A017841 A304329 A111901

Adjacent sequences:  A112189 A112190 A112191 * A112193 A112194 A112195

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 28 2005

STATUS

approved

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Last modified January 16 03:30 EST 2019. Contains 319184 sequences. (Running on oeis4.)