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A092848 Expansion of reciprocal of Hauptmodul for Gamma_0(18). 30
1, -1, 0, 2, -2, -1, 4, -4, -1, 8, -8, -2, 14, -14, -4, 24, -23, -6, 40, -38, -10, 63, -60, -16, 98, -92, -24, 150, -140, -36, 224, -208, -54, 329, -304, -78, 478, -440, -112, 684, -627, -160, 968, -884, -224, 1358, -1236, -312, 1884, -1710, -432, 2592, -2346, -590, 3540, -3196, -801, 4796, -4320, -1082, 6454 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

REFERENCES

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 345 Entry 1(i).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

S. Cooper, Sporadic sequences, modular forms and new series for 1/pi, Ramanujan J. (2012).

W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 155 Eq. (9.13)

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of chi(-q) / chi(-q^3)^3 where chi() is a Ramanujan theta function.

Expansion of q^(-1/3) * c(q^2) / c(q) where c() is a cubic AGM theta function. - Michael Somos, Oct 04 2006

Expansion of q^(-1/3) * eta(q) * eta(q^6)^3 / (eta(q^2) * eta(q^3)^3) in powers of q.

Euler transform of period 6 sequence [-1, 0, 2, 0, -1, 0, ...].

Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + 2*u*v^2.

Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (v^3 - v^2 + v) - u^3 * (1 + 2*v + 4*v^2).

G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A141094. - Michael Somos, Dec 07 2013

G.f.: Product_{k>0} (1 - x^(2*k - 1)) / (1 - x^(6*k - 3))^3.

G.f.: 1 / (1 + (x + x^2) / (1 + (x^2 + x^4) / (1 + (x^3 + x^6) / ...))).

a(n) = A062242(2*n + 1) = (-1)^n * A128111(n). Convolution inverse of A062242.

a(2*n + 1) = - A216046(n). Convolution square is A216046. - Michael Somos, Dec 07 2013

G.f.: T(0), where T(k) = 1 - (x^(k+1)+x^(2*k+2))/((x^(k+1)+x^(2*k+2))+1/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 14 2013

EXAMPLE

G.f. = 1 - x + 2*x^3 - 2*x^4 - x^5 + 4*x^6 - 4*x^7 - x^8 + 8*x^9 + ...

G.f. = q - q^4 + 2*q^10 - 2*q^13 - q^16 + 4*q^19 - 4*q^22 - q^25 + 8*q^28 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 1, n, 2}] / Product[ 1 - x^k, {k, 3, n, 6}]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)

a[ n_] := SeriesCoefficient[  QPochhammer[ x, x^2] / QPochhammer[ x^3, x^6]^3, {x, 0, n}]; (* Michael Somos, Dec 07 2013 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^3 / (eta(x^2 + A) * eta(x^3 + A)^3), n))}; /* Michael Somos, Dec 07 2013 */

(PARI) {a(n) = my(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A + (x*A^2)^2) - x*A^2); polcoeff(A, n))};

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=0, (n-1)\2, (1 - x^(2*k + 1))^if(k%3==1, -2, 1), 1 + x * O(x^n)), n))};

CROSSREFS

Cf. A062242, A128111, A141094, A216046.

Sequence in context: A138189 A110090 A196831 * A128111 A107356 A124725

Adjacent sequences:  A092845 A092846 A092847 * A092849 A092850 A092851

KEYWORD

sign

AUTHOR

Michael Somos, Mar 07 2004

STATUS

approved

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Last modified October 15 18:26 EDT 2019. Contains 328037 sequences. (Running on oeis4.)