A092877 Expansion of (eta(q^4) / eta(q))^8 in powers of q.

1, 8, 44, 192, 718, 2400, 7352, 20992, 56549, 145008, 356388, 844032, 1934534, 4306368, 9337704, 19771392, 40965362, 83207976, 165944732, 325393024, 628092832, 1194744096, 2241688744, 4152367104, 7599231223, 13749863984

Offset: 1

Author

Michael Somos, Mar 19 2004

Keywords

• nonn

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = q + 8*q^2 + 44*q^3 + 192*q^4 + 718*q^5 + 2400*q^6 + 7352*q^7 + 20992*q^8 + ...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A001935, A001936, A005798, A014103, A093160, A124972.

Programs

Mathematica

a[ n_] := If[ n < 0, 0, SeriesCoefficient[ -InverseEllipticNomeQ[ -x] / 16, {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ With[ {lambda = ModularLambda[ Log[x] / ( Pi I)]}, lambda / (16 * (1 - lambda))], {x, 0, n}]]; (* Michael Somos, Jun 13 2011 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^4] / QPochhammer[ q])^8, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
a[1] = 1; a[n_] := a[n] = (8/(n-1))*Sum[DivisorSum[k, Identity, Mod[#, 4] != 0&]*a[n-k], {k, 1, n-1}]; Array[a, 26] (* Jean-François Alcover, Mar 01 2018, after Seiichi Manyama *)
eta[q_]:= q^(1/6) QPochhammer[q]; a[n_]:=SeriesCoefficient[(eta[q^4] / eta[q])^8, {q, 0, n}]; Table[a[n], {n, 4, 35}] (* Vincenzo Librandi, Oct 18 2018 *)

PARI

{a(n) = if( n<0, 0, polcoeff( x * prod(k=1, (n+1)\2, (1 + x^(2*k)) / (1 - x^(2*k-1)), 1 + x * O(x^n))^8, n))};

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^4 + A) / eta(x + A))^8, n))};

Formula

Expansion of q * (psi(-q) / phi(-q))^8 = q * (psi(q^2) / psi(-q))^8 = q * (psi(q) / phi(-q^2))^8 = q * (psi(q^2) / phi(-q))^4 = q * (chi(q) / chi(-q^2)^2)^8 = q / (chi(-q) * chi(-q^2))^8 = q / (chi(q) * chi(-q)^2)^8 = q * (f(-q^4) / f(-q))^8 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jun 13 2011

Euler transform of period 4 sequence [ 8, 8, 8, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^2 - v - 16*u*v - 16*v^2 - 256*u*v^2.

G.f.: x * (Product_{k>0} (1 + x^(2*k)) / (1 - x^(2*k - 1)))^8.

G.f.: theta_2^4 / (16*theta_4^4) = lambda / (16 * (1 - lambda)).

G.f.: exp( Integral theta_3(x)^4/x dx ). - Paul D. Hanna, May 03 2010

a(n) = (-1)^n * A005798(n).

a(2*n) = 8 * A014103(n). - Michael Somos, Aug 09 2015

Convolution inverse of A124972, 8th power of A001935, 4th power of A001936, square of A093160. - Michael Somos, Aug 09 2015

a(n) ~ exp(2*Pi*sqrt(n))/(512*n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(1) = 1, a(n) = (8/(n-1))*Sum_{k=1..n-1} A046897(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017