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A246954 Expansion of phi(-x) * psi(-x)^2 in powers of x where phi(), psi() are Ramanujan theta functions. 3
1, -4, 5, -4, 8, -8, 5, -12, 8, -4, 16, -12, 9, -12, 8, -12, 16, -16, 8, -16, 17, -8, 24, -8, 8, -28, 16, -12, 16, -20, 13, -24, 24, -8, 16, -16, 16, -28, 24, -12, 32, -16, 13, -28, 8, -20, 32, -32, 8, -20, 24, -16, 40, -16, 16, -32, 25, -20, 24, -24, 24, -28 (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..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

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

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 16 * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A246953.

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

a(n) = (-1)^n * A045834(n). a(2*n) = A213022(n). a(2*n + 1) = - 4 * A045831(n).

EXAMPLE

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

G.f. = q - 4*q^5 + 5*q^9 - 4*q^13 + 8*q^17 - 8*q^21 + 5*q^25 - 12*q^29 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x]^2 EllipticTheta[ 2, 0, x]/(2 x^(1/4)), {x, 0, n}]; Table[a[n], {n, 0, 60}]

PROG

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

CROSSREFS

Cf. A045831, A045834, A246953.

Sequence in context: A248671 A232635 A201296 * A045834 A106148 A192038

Adjacent sequences:  A246951 A246952 A246953 * A246955 A246956 A246957

KEYWORD

sign

AUTHOR

Michael Somos, Sep 08 2014

STATUS

approved

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Last modified January 19 12:36 EST 2020. Contains 331049 sequences. (Running on oeis4.)