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

Author

Michael Somos, Oct 04 2006

Keywords

• nonn

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

Programs

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 *)

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))}

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