A002512 Expansion of chi(x)^10 / phi(x)^4 in powers of x where phi(), chi() are Ramanujan theta functions. (Formerly M1380 N0539)

1, 2, 5, 10, 22, 40, 75, 130, 230, 382, 636, 1022, 1645, 2570, 4002, 6110, 9297, 13910, 20715, 30462, 44597, 64584, 93085, 132990, 189164, 266992, 375192, 523800, 728285, 1006684, 1386043, 1898586, 2591120, 3519840, 4764736, 6423032

Offset

0,2

Comments

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

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

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

Morris Newman, Construction and application of a class of modular functions (II). Proc. London Math. Soc. (3) 9 1959 373-387.

Morris Newman, Construction and application of a class of modular functions, II, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]

Formula

G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(4*k)))^-2.

Expansion of 1 / (chi(x)^2 * psi(-x)^4) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Apr 24 2014

Expansion of phi(x) / f(-x^2)^5 in powers of x where phi(),f() are Ramanujan theta functions. - Michael Somos, Apr 24 2014

Expansion of q^(5/12) / (eta(q) * eta(q^4))^2 in powers of q.

Euler transform of period 4 sequence [2, 2, 2, 4, ...]. - Michael Somos, Nov 10 2005

a(n) ~ exp(Pi*sqrt(5*n/3)) * 5^(5/4) / (2^(7/2) * 3^(5/4) * n^(7/4)). - Vaclav Kotesovec, Sep 07 2015

Example

G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 22*x^4 + 40*x^5 + 75*x^6 + 130*x^7 + ...
G.f. = q^-5 + 2*q^7 + 5*q^19 + 10*q^31 + 22*q^43 + 40*q^55 + 75*q^67 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^4])^-2, {x, 0, n}]; (* Michael Somos, Apr 24 2014 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^2]^5, {x, 0, n}]; (* Michael Somos, Apr 24 2014 *)
nmax = 40; CoefficientList[Series[Product[1/((1 - x^k) * (1 - x^(4*k)))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)

Prog

(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^-2, n))}; /* Michael Somos, Nov 10 2005 */

Crossrefs

Cf. A000122, A000700, A010054, A121373.

Sequence in context: A079342 A034456 A272080 * A097096 A073777 A215422

Adjacent sequences: A002509 A002510 A002511 * A002513 A002514 A002515

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Status

approved