A137677 - OEIS

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, 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

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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).

Michael 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: A357069 A033461 A143432 * A015818 A225869 A039972

Adjacent sequences: A137674 A137675 A137676 * A137678 A137679 A137680

KEYWORD

nonn

AUTHOR

Michael Somos, Feb 04 2008

STATUS

approved