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A122830 Expansion of c(q) * c(q^6) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function. 3
1, 1, 0, -2, -3, 0, 5, 7, 0, -12, -15, 0, 26, 32, 0, -50, -63, 0, 92, 114, 0, -168, -201, 0, 295, 350, 0, -496, -591, 0, 818, 967, 0, -1332, -1554, 0, 2126, 2468, 0, -3324, -3855, 0, 5126, 5916, 0, -7824, -8970, 0, 11793, 13471, 0, -17548, -20007, 0, 25857, 29384, 0, -37788, -42771, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000

FORMULA

Expansion of eta(q^2)^2 * eta(q^3)^3 * eta(q^18)^3 / (eta(q) * eta(q^6)^7) in powers of q.

Euler transform of period 18 sequence [ 1, -1, -2, -1, 1, 3, 1, -1, -2, -1, 1, 3, 1, -1, -2, -1, 1, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = 4*v - 4*u^2 + 4*v^2 + 2*w^2 + 8*u*v + 8*v*w + 18*u*v*w + 3*u*w^2 - 12*u^2*w - 12*u^2*v + 6*v^2*w - 3*v^3 - 9*u^2*w^2 - 18*u^2*v*w - 9*u*v^2*w - 9*u^2*v^2 - 9*v^3*w.

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = 2*u6*u1 - 2*u3*u2 + u6*u2^2 - 3*u6*u3*u2 - 3*u3^2*u2 - 4*u3*u2^2 - 3*u3^2*u2^2 + 6*u3^2*u1 - 4*u3*u2*u1 + 4*u6*u2*u1 + u6*u1^2 + 2*u3*u1^2 + 6*u3^2*u1^2 + 3*u6*u3*u1^2 - 6*u6*u3*u2^2 + 6*u3^2*u2*u1 - 6*u6*u3*u2*u1.

Convolution inverse is A182033. - Michael Somos, Feb 19 2015

a(3*n) = 0. - Michael Somos, Feb 19 2015

EXAMPLE

G.f. = q + q^2 - 2*q^4 - 3*q^5 + 5*q^7 + 7*q^8 - 12*q^10 - 15*q^11 + ...

MATHEMATICA

eta[x_] := x^(1/24)*QPochhammer[x]; A122830[n_] := SeriesCoefficient[

eta[q^2]^2* eta[q^3]^3*eta[q^18]^3/(eta[q]*eta[q^6]^7 ), {q, 0, n}]; Table[A122830[n], {n, 0, 50}] (* G. C. Greubel, Aug 11 2017 *)

PROG

(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^3 * eta(x^18 + A)^3 / (eta(x + A) * eta(x^6 + A)^7), n))};

CROSSREFS

Cf. A182033.

Sequence in context: A291305 A004179 A324640 * A321296 A190902 A115562

Adjacent sequences:  A122827 A122828 A122829 * A122831 A122832 A122833

KEYWORD

sign

AUTHOR

Michael Somos, Sep 12 2006

STATUS

approved

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Last modified April 17 06:29 EDT 2021. Contains 343059 sequences. (Running on oeis4.)