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A131964 Expansion of f(x^2, x^10) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function. 8
1, 1, 1, 2, 0, 1, 1, 0, 1, 0, 2, 1, 1, 1, 0, 1, 2, 2, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1, 2, 1, 1, 0, 1, 1, 1, 3, 0, 0, 0, 2, 1, 1, 2, 1, 2, 1, 0, 0, 0, 2, 1, 0, 2, 0, 2, 0, 0, 1, 1, 0, 1, 0, 1, 2, 1, 2, 1, 1, 1, 1, 0, 0, 0, 2, 1, 2, 0, 2, 2, 1, 1, 0, 0, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Expansion of phi(-x^4) * psi(-x^6) / chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

Expansion of q^(-19/24) * eta(q^2) * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [ 1, 0, 1, -2, 1, -1, 1, -1, 1, 0, 1, -2, 1, 0, 1, -1, 1, -1, 1, -2, 1, 0, 1, -2, ...].

a(25*n + 19) = a(n). a(25*n + 4) = a(25*n + 9) = a(25*n + 14) = a(25*n + 24) = 0.

2 * a(n) = A123484(24*n + 19).

EXAMPLE

G.f. = 1 + x + x^2 + 2*x^3 + x^5 + x^6 + x^8 + 2*x^10 + x^11 + x^12 + x^13 + ...

G.f. = q^19 + q^43 + q^67 + 2*q^91 + q^139 + q^163 + q^211 + 2*q^259 + q^283 + ...

MATHEMATICA

a[ n_] := If[ n < 0, 0, With[ {m = 24 n + 19}, DivisorSum[ m, KroneckerSymbol[ -12, #] Mod[m/#, 2] &] / 2]]; (* Michael Somos, Nov 03 2015 *)

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) EllipticTheta[ 4, 0, x^4] QPochhammer[ -x, x] EllipticTheta[ 2, Pi/4, x^3], {x, 0, n}]; (* Michael Somos, Nov 03 2015 *)

PROG

(PARI) {a(n) = if( n<0, 0, n = 24*n + 19; sumdiv(n, d, kronecker( -12, d) * (n/d %2)) / 2)};

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)), n))};

CROSSREFS

Cf. A123484.

Sequence in context: A091586 A330667 A116377 * A091430 A260728 A065339

Adjacent sequences:  A131961 A131962 A131963 * A131965 A131966 A131967

KEYWORD

nonn

AUTHOR

Michael Somos, Aug 02 2007

STATUS

approved

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Last modified June 19 05:14 EDT 2021. Contains 345125 sequences. (Running on oeis4.)