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A112206
Coefficients of replicable function number "72b".
9
1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 4, 4, 7, 7, 6, 10, 11, 11, 14, 16, 17, 21, 22, 24, 32, 34, 34, 44, 49, 50, 60, 66, 72, 84, 90, 98, 117, 125, 132, 156, 171, 181, 206, 226, 245, 277, 298, 322, 369, 397, 422, 480, 522, 557, 620, 674, 728, 807, 868, 936, 1043, 1121, 1198
OFFSET
0,4
COMMENTS
From Michael Somos, Oct 28 2019: (Start)
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution squared is A112173.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
Given G.f. A(x), then B(q) = q^(-1) * A(q^6) satisifes 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + (u^2 - v)*v*w^2 + (u^2 + v)*v^2.
(End)
LINKS
D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
Expansion of q^(1/6)*((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))) in powers of q. - G. C. Greubel, Jun 01 2018
From Michael Somos, Oct 28 2019: (Start)
Expansion of chi(x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.
Euler transform of period 12 sequence [1, -1, 2, 0, 1, -2, 1, 0, 2, -1, 1, 0, ...].
G.f.: Product_{k>=0} (1 + x^(2*k + 1)) * (1 + x^(6*k + 3)).
a(n) = (-1)^n * A112175(n). a(2*n) = A328789(n). a(2*n + 1) = A328790(n).
(End)
EXAMPLE
G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
G.f. = q^-1 + q^5 + 2*q^17 + 2*q^23 + q^29 + 2*q^35 + 2*q^41 + ...
MATHEMATICA
nmax = 60; CoefficientList[Series[Product[(1 + x^k)*(1 + x^(3*k)) / ((1 + x^(2*k))*(1 + x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; h:= q^(1/6)*((eta[q^2]*eta[q^6])^2/(eta[q]*eta[q^3]*eta[q^4]*eta[q^12])); a:= CoefficientList[Series [h, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 01 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3, x^6], {x, 0 , n}]; (* Michael Somos, Oct 28 2019 *)
PROG
(PARI) q='q+O('q^50); h=((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))); Vec(h) \\ G. C. Greubel, Jun 01 2018
(PARI) {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))}; /* Michael Somos, Oct 28 2019 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Aug 28 2005
STATUS
approved