OFFSET
0,2
COMMENTS
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
KEYWORD
nonn
AUTHOR
STATUS
approved