A259393 Expansion of f(-x^3, -x^7) * f(x^4, x^6) / psi(-x)^2 in powers of x where psi(), f(,) are Ramanujan theta functions.

1, 2, 3, 5, 10, 17, 26, 39, 61, 93, 133, 189, 272, 385, 530, 725, 994, 1349, 1803, 2395, 3180, 4194, 5476, 7115, 9230, 11912, 15266, 19490, 24828, 31500, 39766, 50034, 62813, 78600, 97970, 121769, 151032, 186809, 230340, 283326, 347811, 425964, 520336, 634260

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, 0, 1, 3, 2, 1, 1, 1, 2, -2, 2, 1, 1, 1, 2, 3, 1, 0, 2, 0, ...].

a(n) = - A132225(5*n) unless n=0.

a(n) ~ exp(2*Pi*sqrt(n/5)) * phi / (2 * 5^(5/4) * n^(3/4)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 04 2018

Example

G.f. = 1 + 2*x + 3*x^2 + 5*x^3 + 10*x^4 + 17*x^5 + 26*x^6 + 39*x^7 + ...

Mathematica

QP := QPochhammer; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]*QP[x*y, x*y]; A:= f[-x^3, -x^7]*f[x^4, x^6]/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, 0, -1, -3, -2, -1, -1, -1, -2, 2, -2, -1, -1, -1, -2, -3, -1, 0, -2] [k%20 + 1]), n))};

Crossrefs

Cf. A132225.

Sequence in context: A364641 A047011 A359518 * A018163 A339293 A195507

Adjacent sequences: A259390 A259391 A259392 * A259394 A259395 A259396

Keyword

nonn

Author

Michael Somos, Jun 25 2015

Status

approved