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A246926 Expansion of phi(x)^2 * chi(x) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions. 3
1, 5, 8, 4, 4, 13, 12, 4, 5, 16, 24, 8, 4, 20, 12, 8, 9, 20, 32, 4, 12, 29, 12, 8, 8, 36, 40, 8, 8, 20, 24, 16, 8, 25, 40, 12, 12, 32, 24, 12, 13, 48, 40, 8, 8, 40, 36, 8, 16, 20, 56, 16, 12, 52, 12, 20, 13, 36, 56, 16, 20, 40, 24, 8, 8, 45, 72, 12, 16, 52 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of q^(-1/3) * eta(q^2)^12 * eta(q^3) * eta(q^12) / (eta(q)^5 * eta(q^4)^5 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [5, -7, 4, -2, 5, -7, 5, -2, 4, -7, 5, -3, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 72^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246927.

2 * a(n) = A246928(3*n + 1).

EXAMPLE

G.f. = 1 + 5*x + 8*x^2 + 4*x^3 + 4*x^4 + 13*x^5 + 12*x^6 + 4*x^7 + 5*x^8 + ...

G.f. = q + 5*q^4 + 8*q^7 + 4*q^10 + 4*q^13 + 13*q^16 + 12*q^19 + 4*q^22 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}]; (* Michael Somos, Jan 08 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^12 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^5 * eta(x^4 + A)^5 * eta(x^6 + A)), n))};

(MAGMA) A := Basis( ModularForms( Gamma0(36), 3/2), 210); A[2] + 5*A[5];

CROSSREFS

Cf. A246927, A246928.

Sequence in context: A199450 A019649 A224833 * A199288 A099878 A167901

Adjacent sequences:  A246923 A246924 A246925 * A246927 A246928 A246929

KEYWORD

nonn

AUTHOR

Michael Somos, Sep 07 2014

STATUS

approved

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Last modified February 22 17:14 EST 2020. Contains 332140 sequences. (Running on oeis4.)