A112192 - OEIS

%I #28 Mar 12 2021 22:24:43

%S 1,1,2,2,3,4,5,7,8,10,13,16,20,24,30,36,43,52,61,73,86,102,120,140,

%T 165,192,224,260,301,348,401,462,530,608,696,796,909,1035,1178,1338,

%U 1518,1720,1945,2198,2480,2796,3148,3540,3978,4464,5006,5606,6273,7012

%N Coefficients of replicable function number "48h".

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

%H Seiichi Manyama, <a href="/A112192/b112192.txt">Table of n, a(n) for n = 0..1000</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%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>

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F a(n) ~ exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015

%F 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

%F 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

%F Euler transform of period 12 sequence [1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, ...]. - _Michael Somos_, Sep 30 2015

%F a(n) = number of partitions of n into parts == +-1, +-2, +-5 (mod 12). - _Michael Somos_, Sep 30 2015

%F 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

%F a(n) = A112186(2*n) = A112187(2*n). - _Michael Somos_, Sep 30 2015

%F Convolution inverse of A262771. - _Michael Somos_, Sep 30 2015

%e 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 + ...

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

%t 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 *)

%t a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^4] / ( QPochhammer[ x] QPochhammer[ x^12]), {x, 0, n}]; (* _Michael Somos_, Sep 30 2015 *)

%o (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 */

%Y Cf. A058578, A187091.

%Y Cf. A112186, A112187, A262771.

%K nonn

%O 0,3

%A _Michael Somos_, Aug 28 2005