A187428 -id:A187428 - cp's OEIS frontend

A187427 Expansion of q^(3/8) * eta(q)^3 / eta(q^3)^4 in powers of q.

1, -3, 0, 9, -12, 0, 27, -42, 0, 82, -111, 0, 207, -279, 0, 486, -630, 0, 1055, -1362, 0, 2205, -2775, 0, 4374, -5472, 0, 8427, -10389, 0, 15696, -19224, 0, 28539, -34614, 0, 50630, -61059, 0, 88119, -105483, 0, 150417, -179178, 0, 252727, -299325, 0, 418068

Offset: 0

Michael Somos, Mar 09 2011

sign

			1 - 3*x + 9*x^3 - 12*x^4 + 27*x^6 - 42*x^7 + 82*x^9 - 111*x^10 + ...
q^-3 - 3*q^5 + 9*q^21 - 12*q^29 + 27*q^45 - 42*q^53 + 82*q^69 - 111*q^77 + ...

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

Cf. A053762, A187248.

QP = QPochhammer; s = QP[q]^3/QP[q^3]^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(3/8) *eta[q]^3/ eta[q^3]^4, {q, 0, 50}], q] (* G. C. Greubel, Aug 14 2018 *)

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

Euler transform of period 3 sequence [ -3, -3, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = (9/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A187428.
G.f.: Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k))^4.
a(3*n) = A053762(n). a(3*n + 1) = -3 * A187428(n). a(3*n + 2) = 0.

A187429 Expansion of q^(3/8) * a(q) / eta(q^3)^3 in powers of q where a() is a cubic AGM function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 09 2011

nonn

			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 + ...

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.

Cf. A053762, A187428.

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-François Alcover, Nov 06 2015, adapted from 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))}

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.

cp's OEIS Frontend

A187427 Expansion of q^(3/8) * eta(q)^3 / eta(q^3)^4 in powers of q.

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