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A258593 Expansion of (phi(x^2) * psi(x^2) / phi(-x)^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions. 1
1, 8, 46, 208, 805, 2776, 8742, 25584, 70450, 184232, 460832, 1108848, 2578295, 5814992, 12760598, 27317056, 57174768, 117223008, 235818894, 466154416, 906606234, 1736736024, 3280271526, 6114139616, 11255369609, 20478505104, 36849912318, 65619691088 (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

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

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

Expansion of q^(-1/2) * (eta(q^4)^7 / (eta(q)^4 * eta(q^2) * eta(q^8)^2))^2 in powers of q.

Euler transform of period 8 sequence [ 8, 10, 8, -4, 8, 10, 8, 0, ...].

-4 * a(n) = A260186(2*n + 1).

a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

EXAMPLE

G.f. = 1 + 8*x + 46*x^2 + 208*x^3 + 805*x^4 + 2776*x^5 + 8742*x^6 + ...

G.f. = q + 8*q^3 + 46*q^5 + 208*q^7 + 805*q^9 + 2776*q^11 + 8742*q^13 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) (EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, 0, x] / EllipticTheta[ 4, 0 , x]^2)^2, {x, 0, n}];

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

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^7 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^8 + A)^2))^2, n))};

CROSSREFS

Cf. A260186.

Sequence in context: A034469 A212673 A183392 * A134114 A071586 A027650

Adjacent sequences:  A258590 A258591 A258592 * A258594 A258595 A258596

KEYWORD

nonn

AUTHOR

Michael Somos, Nov 06 2015

STATUS

approved

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Last modified February 17 18:46 EST 2019. Contains 320222 sequences. (Running on oeis4.)