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

1,2

Comments

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

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

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

Example

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

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 *)

Prog

(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))};

Crossrefs

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

Sequence in context: A341386 A181358 A005798 * A160521 A277958 A283077

Adjacent sequences: A092874 A092875 A092876 * A092878 A092879 A092880

Keyword

nonn

Author

Michael Somos, Mar 19 2004

Status

approved