A097243 Expansion of 1 + 32 * (eta(q^4) / eta(q))^8 in powers of q.

1, 32, 256, 1408, 6144, 22976, 76800, 235264, 671744, 1809568, 4640256, 11404416, 27009024, 61905088, 137803776, 298806528, 632684544, 1310891584, 2662655232, 5310231424, 10412576768, 20098970624, 38231811072, 71734039808, 132875747328, 243175399136

Offset: 0

Author

Michael Somos, Aug 02 2004

Keywords

• nonn

Comments

Expansion of a q-series used in construction of j(tau) to j(2tau) iteration.

Examples

			G.f. = 1 + 32*x + 256*x^2 + 1408*x^3 + 6144*x^4 + 22976*x^5 + 76800*x^6 + ...
		

References

• H. Cohn, Introduction to the construction of class fields, Cambridge 1985, p. 191

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A007248, A007096, A014969, A092877, A139820, A189925, A212318, A232358.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ 1 + 32 x (QPochhammer[ x^4] / QPochhammer[ x])^8, {x, 0, n}]; (* Michael Somos, Dec 15 2016 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x^n * O(x); polcoeff( 1 + 32 * x * (eta(x^4 + A) / eta(x + A))^8, n))};

Formula

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+3)^2 - 8*(u+1)*v^2.

a(n) = 32*A092877(n), if n>0. a(n) = A007096(4*n).

a(n) = A014969(2*n) = A139820(2*n) = A189925(4*n) = A212318(4*n) = A232358(4*n). - Michael Somos, Dec 15 2016

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 1/8 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A007248. - Michael Somos, Dec 15 2016

a(n) ~ exp(2*Pi*sqrt(n))/(16*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017