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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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: A024369 A306006 A047011 * A018163 A339293 A195507

Adjacent sequences:  A259390 A259391 A259392 * A259394 A259395 A259396

KEYWORD

nonn

AUTHOR

Michael Somos, Jun 25 2015

STATUS

approved

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Last modified June 27 22:27 EDT 2022. Contains 354899 sequences. (Running on oeis4.)