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A128763 Expansion of chi(q^5) * chi(q^10) / ( chi(q) * chi(q^2)) in powers of q where chi() is a Ramanujan theta function. 2
1, -1, 0, -1, 2, -1, 0, -2, 3, -2, 2, -4, 6, -5, 4, -6, 9, -8, 6, -10, 15, -14, 12, -17, 24, -21, 18, -26, 35, -32, 30, -42, 52, -50, 48, -60, 75, -74, 70, -88, 111, -109, 104, -130, 158, -154, 150, -184, 220, -218, 218, -262, 308, -308, 308, -362, 421, -426, 428, -498, 580, -589, 592, -685, 788, -796 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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..1500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

FORMULA

Euler transform of period 40 sequence [ -1, 0, -1, 1, 0, 0, -1, 0, -1, 0, -1, 1, -1, 0, 0, 0, -1, 0, -1, 0, -1, 0, -1, 0, 0, 0, -1, 1, -1, 0, -1, 0, -1, 0, 0, 1, -1, 0, -1, 0, ...].

Given g.f. A(x), then B(q) = 1/q*A(q^2) satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (u^2 + v^2).

G.f.: Product_{k>0} (1 + x^(4*k)) * (1 + x^(5*k)) / ( (1 + x^k) * (1 + x^(20*k)) ).

Convolution inverse of A128762.

Expansion of q^(1/2)*eta(q)*eta(q^8)*eta(q^10)*eta(q^20)/(eta(q^2)* eta(q^4)*eta(q^5)*eta(q^40)) in powers of q. - G. C. Greubel, Jul 03 2018

EXAMPLE

G.f. = 1 - x - x^3 + 2*x^4 - x^5 - 2*x^7 + 3*x^8 - 2*x^9 + 2*x^10 - 4*x^11 + ...

G.f. = 1/q - q - q^5 + 2*q^7 - q^9 - 2*q^13 + 3*q^15 - 2*q^17 + 2*q^19 - ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ x, -x] QPochhammer[ x^2, -x^2]) / (QPochhammer[ x^5, - x^5] QPochhammer[ x^10, -x^10]) , {x, 0, n}]; (* Michael Somos, Apr 26 2015 *)

eta[q_] := q^(1/24)*QPochhammer[q]; A:= q^(1/2)*eta[q]*eta[q^8]* eta[q^10]*eta[q^20]/(eta[q^2]*eta[q^4]*eta[q^5] *eta[q^40]); a:= CoefficientList[Series[A, {q, 0, 80}], q]; Table[a[[n]], {n, 1, 80}] (* G. C. Greubel, Jul 03 2018 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^8 + A) * eta(x^10 + A) * eta(x^20 + A) / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^40 + A)), n))};

CROSSREFS

Cf. A128762.

Sequence in context: A072739 A328699 A030399 * A127597 A167749 A104770

Adjacent sequences: A128760 A128761 A128762 * A128764 A128765 A128766

KEYWORD

sign

AUTHOR

Michael Somos, Mar 25 2007

STATUS

approved

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Last modified January 29 01:41 EST 2023. Contains 359905 sequences. (Running on oeis4.)