A164269 Expansion of q * f(q^9)^3 * phi(q) / (f(q^3) * phi(q^3)^3) in powers of q where f(), phi() are Ramanujan theta functions.

1, 2, 0, -7, -12, 0, 32, 50, 0, -114, -168, 0, 350, 496, 0, -967, -1332, 0, 2468, 3324, 0, -5916, -7824, 0, 13471, 17548, 0, -29384, -37788, 0, 61784, 78578, 0, -125838, -158496, 0, 249230, 311224, 0, -481506, -596676, 0, 909788, 1119624, 0, -1684824, -2060448, 0

Offset

1,2

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 36 sequence [2, -3, -5, -1, 2, 8, 2, -1, -2, -3, 2, 3, 2, -3, -5, -1, 2, 2, 2, -1, -5, -3, 2, 3, 2, -3, -2, -1, 2, 8, 2, -1, -5, -3, 2, 0, ...].

a(3*n) = 0. a(3*n + 1) = A164270(n). a(3*n + 2) = 2 * A164271(n).

Convolution inverse of A164268.

Expansion of x * phi(x) * chi(x^3) * f(x^9)^3 / phi(x^3)^4 = x * phi(x) * f(x^9)^3 / (chi(x^3)^3 * f(x^3)^4) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 20 2017

Example

G.f. = q + 2*q^2 - 7*q^4 - 12*q^5 + 32*q^7 + 50*q^8 - 114*q^10 - 168*q^11 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164269[n_] := SeriesCoefficient[q*(f[q^9, -q^18]^3*f[q, q])/(( f[q^3, -q^6])*f[q^3, q^3]^3), {q, 0, n}]; Rest[Table[A164269[n], {n, 0, 50}]] (* G. C. Greubel, Sep 16 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^9]^3 EllipticTheta[ 3, 0, x] / (QPochhammer[ -x^3] EllipticTheta[ 3, 0, x^3]^3), {x, 0, n}]; (* Michael Somos, Sep 17 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ x^9]^3 EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^3]^4, {x, 0, n}]; /(* Michael Somos, Sep 20 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3), n))};

Crossrefs

Cf. A164268, A164270, A164271.

Sequence in context: A021485 A019821 A016631 * A293265 A121814 A195298

Adjacent sequences: A164266 A164267 A164268 * A164270 A164271 A164272

Keyword

sign

Author

Michael Somos, Aug 11 2009

Status

approved