A002512 - OEIS

%I M1380 N0539 #32 Mar 08 2018 02:50:45

%S 1,2,5,10,22,40,75,130,230,382,636,1022,1645,2570,4002,6110,9297,

%T 13910,20715,30462,44597,64584,93085,132990,189164,266992,375192,

%U 523800,728285,1006684,1386043,1898586,2591120,3519840,4764736,6423032

%N Expansion of chi(x)^10 / phi(x)^4 in powers of x where phi(), chi() are Ramanujan theta functions.

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

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

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

%H G. C. Greubel, <a href="/A002512/b002512.txt">Table of n, a(n) for n = 0..1000</a>

%H Morris Newman, <a href="https://doi.org/10.1112/plms/s3-9.3.373">Construction and application of a class of modular functions (II)</a>. Proc. London Math. Soc. (3) 9 1959 373-387.

%H Morris Newman, <a href="/A002507/a002507.pdf">Construction and application of a class of modular functions, II</a>, Proc. London Math. Soc. (3) 9 1959 373-387. [Annotated scanned copy, barely legible]

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

%F 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

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

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

%F Euler transform of period 4 sequence [2, 2, 2, 4, ...]. - _Michael Somos_, Nov 10 2005

%F 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

%e G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 22*x^4 + 40*x^5 + 75*x^6 + 130*x^7 + ...

%e G.f. = q^-5 + 2*q^7 + 5*q^19 + 10*q^31 + 22*q^43 + 40*q^55 + 75*q^67 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^4])^-2, {x, 0, n}]; (* _Michael Somos_, Apr 24 2014 *)

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] / QPochhammer[ x^2]^5, {x, 0, n}]; (* _Michael Somos_, Apr 24 2014 *)

%t 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 *)

%o (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 */

%Y Cf. A000122, A000700, A010054, A121373.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, _Simon Plouffe_