A116494 Expansion of psi(q^5)/psi(q) in powers of q where psi() is a Ramanujan theta function.

1, -1, 1, -2, 3, -3, 4, -6, 8, -10, 12, -16, 21, -25, 30, -38, 48, -57, 68, -84, 102, -121, 143, -172, 207, -243, 284, -338, 400, -465, 542, -636, 744, -862, 996, -1158, 1344, -1546, 1776, -2050, 2361, -2701, 3088, -3540, 4050, -4613, 5248, -5980, 6808, -7719, 8742, -9916, 11232, -12682

Offset

0,4

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).

Links

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

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2)*eta(q)*eta(q^10)^2/(eta(q^2)^2*eta(q^5)) in powers of q.

Euler transform of period 10 sequence [ -1,1,-1,1,0,1,-1,1,-1,0,...].

Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2)) where f(u,v)=(1-u^2)(1-5u^2)v^2 -(u^2-v^2)^2.

Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2),B(x^4)) where f(u,v,w)=v*w*(1-v^2)-u^2*(v+w)^2.

Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2),B(x^3),B(x^6)) where f(u1,u2,u3,u6)=u2*u6*(u1^2-u3^2) -(u2*u3-u1*u6)^2.

G.f.: Product_{k>0} (1-x^k)/(1-x^(5k))*((1-x^(10k))/(1-x^(2k)))^2 = (Sum_{k>0} x^(5(k^2-k)/2))/(Sum_{k>0} x^((k^2-k)/2)).

a(n) = (-1)^n*A036026(n).

Mathematica

a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(5/2)]/EllipticTheta[2, 0, q^(1/2)]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^10+A)^2/eta(x^2+A)^2/eta(x^5+A), n))}

Crossrefs

Cf. A036026.

Sequence in context: A036030 A036022 A036026 * A036031 A218947 A017818

Adjacent sequences: A116491 A116492 A116493 * A116495 A116496 A116497

Keyword

sign

Author

Michael Somos, Feb 18 2006

Status

approved