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A137676 Expansion of f(-x^2, -x^3) / f(-x, -x^3) in powers of x where f(, ) is Ramanujan's general theta function. 1
1, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 2, 2, 0, 0, 3, 3, 0, 0, 4, 4, 0, 0, 5, 6, 0, 0, 7, 7, 0, 0, 9, 10, 0, 0, 12, 12, 0, 0, 15, 16, 0, 0, 19, 20, 0, 0, 24, 26, 0, 0, 30, 31, 0, 0, 37, 40, 0, 0, 46, 48, 0, 0, 57, 60, 0, 0, 69, 72, 0, 0, 84, 89, 0, 0, 102, 106, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,10

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

G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 36, Eq. (4.11). MR0858826 (88b:11063).

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of f(-x^2) * f(-x^5) / (f(-x^4) * f(-x, -x^4)) in powers of x where f(, ) is Ramanujan's general theta function.

Expansion of (f(-x^13, -x^17) + x * f(-x^7, -x^23)) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Euler transform of period 20 sequence [ 1, -1, 0, 1, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, 0, 1, 0, -1, 1, 0, ...].

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

a(4*n) = A122129(n). a(4*n + 1) = A122135(n). a(4*n + 2) = a(4*n + 3) = 0.

G.f.: (Sum_{k in Z} (-1)^k^2 * x^(k * (5*k + 1) / 2)) / (Sum_{k in Z} (-1)^k^2 * x^(k * (2*k + 1))). - Michael Somos, Oct 08 2015

EXAMPLE

G.f. = 1 + x + x^4 + x^5 + x^8 + 2*x^9 + 2*x^12 + 2*x^13 + 3*x^16 + 3*x^17 + ...

G.f. = 1/q + q^9 + q^39 + q^49 + q^79 + 2*q^89 + 2*q^119 + 2*q^129 + 3*q^159 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] / (QPochhammer[ x^4] QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(k^2) / QPochhammer[ x^4, x^4, k], {k, 0, Sqrt @ n}], {x, 0, n}]]; (* Michael Somos, Oct 08 2015 *)

a[ n_] := SeriesCoefficient[ Sqrt[2] x^(1/8) QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5] QPochhammer[ x^5] / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)

PROG

(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n), x^k^2 / prod(i=1, k, 1 - x^(4*i), 1 + x * O(x^(n - k^2)))), n))};

CROSSREFS

Cf. A122129, A122135.

Sequence in context: A194947 A132339 A333941 * A333755 A238130 A238707

Adjacent sequences:  A137673 A137674 A137675 * A137677 A137678 A137679

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 04 2008

STATUS

approved

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Last modified October 26 17:15 EDT 2020. Contains 338027 sequences. (Running on oeis4.)