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A187429 Expansion of q^(3/8) * a(q) / eta(q^3)^3 in powers of q where a() is a cubic AGM function. 1
1, 6, 0, 9, 24, 0, 27, 84, 0, 82, 222, 0, 207, 558, 0, 486, 1260, 0, 1055, 2724, 0, 2205, 5550, 0, 4374, 10944, 0, 8427, 20778, 0, 15696, 38448, 0, 28539, 69228, 0, 50630, 122118, 0, 88119, 210966, 0, 150417, 358356, 0, 252727, 598650, 0, 418068, 986022 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..2500

J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

FORMULA

Expansion of q^(3/8) * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^4 in powers of q.

G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = (3/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A053762.

a(3*n) = A053762(n). a(3*n+ 1) = 6 * A187428(n). a(3*n + 2) = 0.

EXAMPLE

G.f. = 1 + 6*x + 9*x^3 + 24*x^4 + 27*x^6 + 84*x^7 + 82*x^9 + 222*x^10 + ...

G.f. = q^-3 + 6*q^5 + 9*q^21 + 24*q^29 + 27*q^45 + 84*q^53 + 82*q^69 + 222*q^77 + ...

MATHEMATICA

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x + A]^3 + 9*x*QPochhammer[x^9 + A]^3)/QPochhammer[x^3 + A]^4, {x, 0, n}]]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Nov 06 2015, adapted from PARI *)

PROG

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / eta(x^3 + A)^4, n))}

CROSSREFS

Cf. A053762, A187428.

Sequence in context: A176403 A153314 A019622 * A083573 A117006 A294345

Adjacent sequences:  A187426 A187427 A187428 * A187430 A187431 A187432

KEYWORD

nonn

AUTHOR

Michael Somos, Mar 09 2011

STATUS

approved

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Last modified April 10 21:21 EDT 2021. Contains 342856 sequences. (Running on oeis4.)