A113261 Expansion of (9*phi(q)*phi(q^3)^5 - phi(q)^5*phi(q^3))/8 in powers of q where phi(q) is a Ramanujan theta function.

1, 1, -5, 1, 11, -24, -5, 50, -53, 1, 120, -120, 11, 170, -250, -24, 203, -288, -5, 362, -264, 50, 600, -528, -53, 601, -850, 1, 550, -840, 120, 962, -821, -120, 1440, -1200, 11, 1370, -1810, 170, 1272, -1680, -250, 1850, -1320, -24, 2640, -2208, 203, 2451, -3005, -288, 1870, -2808

Offset

0,3

Comments

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

References

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 226 Entry 4(ii).

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

Euler transform of period 12 sequence [1, -6, 6, -5, 1, -12, 1, -5, 6, -6, 1, -6, ...].

Expansion of eta(q^2)^7 * eta(q^6)^11 / (eta(q) * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.

a(n) is multiplicative and a(3^e) = 1, a(2^e) = ((-4)^(e+1)-1)/(-4-1)-2, a(p^e) = ((-p^2)^(e+1)-1)/(-p-1) if p == 2 (mod 3), a(p^e) = ((p^2)(e+1)-1)/(p-1) if p == 1 (mod 3).

G.f.: 1 + Sum_{k>0} k^2 x^k/(1-(-x)^k) Kronecker(-3, k).

Example

G.f. = 1 + q - 5*q^2 + q^3 + 11*q^4 - 24*q^5 - 5*q^6 + 50*q^7 + ... - Michael Somos, Sep 07 2018

Mathematica

a[n_]:= SeriesCoefficient[(9*EllipticTheta[3, 0, q]*EllipticTheta[3, 0, q^3]^5 - EllipticTheta[3, 0, q]^5*EllipticTheta[3, 0, q^3])/8, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Feb 09 2018 *)

Prog

(PARI) {a(n) = if( n<1, n==0, sumdiv(n, d, d^2 * kronecker(-3, d) * (-1)^(n-d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^6 + A)^11 / (eta(x + A) * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};
(PARI) {a(n) = my(A, p, e, t); if( n<1, n==0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 1, t=p^2*(-1)^(p%3==2); (t^(e+1) - 1) / (t-1) - 2*(p==2))))};

Crossrefs

Sequence in context: A332733 A132000 A132001 * A063004 A351573 A347135

Adjacent sequences: A113258 A113259 A113260 * A113262 A113263 A113264

Keyword

sign,mult

Author

Michael Somos, Oct 21 2005

Status

approved