A137828 - OEIS

A137828

Expansion of phi(x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

1, 2, 0, 0, 4, 4, 0, 0, 9, 12, 0, 0, 20, 24, 0, 0, 42, 50, 0, 0, 80, 92, 0, 0, 147, 172, 0, 0, 260, 296, 0, 0, 445, 510, 0, 0, 744, 840, 0, 0, 1215, 1372, 0, 0, 1944, 2176, 0, 0, 3059, 3424, 0, 0, 4740, 5268, 0, 0, 7239, 8040, 0, 0, 10920, 12072, 0, 0, 16286

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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..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Euler transform of period 4 sequence [ 2, -3, 2, 1, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 6^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A051136.

G.f.: Product_{k>0} (1 + x^k) / ( (1 - x^k) * (1 + x^(2*k))^4 ).

a(4*n + 2) = a(4*n + 3) = 0.

a(4*n) = A051136(n). a(4*n + 1) = 2 * A137829(n).

EXAMPLE

G.f. = 1 + 2*x + 4*x^4 + 4*x^5 + 9*x^8 + 12*x^9 + 20*x^12 + 24*x^13 + 42*x^16 + ...

G.f. = 1/q + 2*q^2 + 4*q^11 + 4*q^14 + 9*q^23 + 12*q^26 + 20*q^35 + 24*q^38 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)

PROG

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

CROSSREFS

Cf. A051136, A137829.

Sequence in context: A182056 A072070 A137830 * A264655 A264476 A342276

Adjacent sequences: A137825 A137826 A137827 * A137829 A137830 A137831

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 12 2008

STATUS

approved