A259392 Expansion of phi(-x^5) * f(x^2, x^8) / psi(-x)^2 in powers of x where phi, psi, f(,) are Ramanujan theta functions.

1, 2, 4, 8, 14, 22, 35, 54, 82, 122, 176, 254, 362, 504, 697, 960, 1307, 1762, 2360, 3142, 4158, 5462, 7133, 9280, 12013, 15462, 19818, 25314, 32198, 40782, 51476, 64768, 81226, 101522, 126502, 157216, 194846, 240784, 296802, 365006, 447794, 548042, 669254

Offset

0,2

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 = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 20 sequence [ 2, 1, 2, 1, 0, 0, 2, 3, 2, -2, 2, 3, 2, 0, 0, 1, 2, 1, 2, 0, ...].

a(n) = A132225(5*n + 1).

a(n) ~ exp(2*Pi*sqrt(n/5)) / (5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jul 04 2018

Example

G.f. = 1 + 2*x + 4*x^2 + 8*x^3 + 14*x^4 + 22*x^5 + 35*x^6 + 54*x^7 + ...
G.f. = q + 2*q^6 + 4*q^11 + 8*q^16 + 14*q^21 + 22*q^26 + 35*q^31 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2, x^10] QPochhammer[ -x^8, x^10] (QPochhammer[ x^2, x^4] QPochhammer[ x^5] / QPochhammer[ x])^2, {x, 0, n}];
QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; A:= f[-x^5, -x^5]*f[x^2, x^8]/f[-x, - x^3]^2; a:= CoefficientList[Series[A, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 04 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, -2, -1, -2, -1, 0, 0, -2, -3, -2, 2, -2, -3, -2, 0, 0, -1, -2, -1, -2][k%20 + 1]), n))};

Crossrefs

Cf. A132225.

Sequence in context: A053798 A305497 A231429 * A261968 A138526 A286522

Adjacent sequences: A259389 A259390 A259391 * A259393 A259394 A259395

Keyword

nonn

Author

Michael Somos, Jun 25 2015

Status

approved