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A137677 Expansion of f(-x, -x^4) / psi(-x) where psi() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function. 1
1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 3, 4, 0, 0, 4, 4, 0, 0, 5, 6, 0, 0, 7, 7, 0, 0, 9, 10, 0, 0, 11, 11, 0, 0, 14, 16, 0, 0, 18, 18, 0, 0, 22, 24, 0, 0, 27, 28, 0, 0, 34, 36, 0, 0, 41, 42, 0, 0, 50, 54, 0, 0, 61, 62, 0, 0, 73 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,16

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, Equ. (4.12). 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^2, -x^3) ) in powers of x where f(, ) is Ramanujan's general theta function.

Expansion of ( f(-x^11, -x^19) + x^3 * f(-x, -x^29) ) / f(-x^4) in powers of x where f(, ) is Ramanujan's general theta function.

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

a(4*n + 1) = a(4*n + 2) = 0. a(4*n) = A122134(n). a(4*n + 3) = A122130(n).

EXAMPLE

G.f. = 1 + x^3 + x^7 + x^8 + x^11 + x^12 + 2*x^15 + 2*x^16 + 2*x^19 + 2*x^20 + ...

G.f. = q + q^31 + q^71 + q^81 + q^111 + q^121 + 2*q^151 + 2*q^161 + 2*q^191 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^4] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)

PROG

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

CROSSREFS

Cf. A122130, A122134.

Sequence in context: A321447 A033461 A143432 * A015818 A225869 A039972

Adjacent sequences:  A137674 A137675 A137676 * A137678 A137679 A137680

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 04 2008

STATUS

approved

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Last modified October 20 10:00 EDT 2020. Contains 337900 sequences. (Running on oeis4.)