A123653 Expansion of (eta(q^2)eta(q^6)/(eta(q)eta(q^3)))^6 in powers of q.

1, 6, 21, 68, 198, 510, 1248, 2904, 6393, 13604, 28044, 55956, 108982, 207552, 386622, 707216, 1271970, 2250582, 3925780, 6757272, 11483232, 19290824, 32057352, 52722744, 85884503, 138644292, 221885805, 352241792, 554892894

Offset

1,2

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Expansion of q/(chi(-q)*chi(-q^3))^6 in powers of q where chi() is a Ramanujan theta function.

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Euler transform of period 6 sequence [ 6, 0, 12, 0, 6, 0, ...].

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= u^2 -v*(1+12*u+64*u*v)

G.f.: x*(Product_{k>0} (1+x^k)*(1+x^(3k)))^6.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (64 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Convolution inverse of A121666. - Seiichi Manyama, Mar 30 2017

Mathematica

nmax = 40; Rest[CoefficientList[Series[x * Product[((1+x^k) * (1+x^(3*k)))^6, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

Prog

(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^6+A)/eta(x+A)/eta(x^3+A))^6, n))}

Crossrefs

Sequence in context: A180795 A306089 A107653 * A364636 A318943 A200761

Adjacent sequences: A123650 A123651 A123652 * A123654 A123655 A123656

Keyword

nonn

Author

Michael Somos, Oct 04 2006

Status

approved