A163819 Expansion of (phi(q) * phi(q^10) - phi(q^2) * phi(q^5)) / 2 in powers of q where phi() is a Ramanujan theta function.

1, -1, 0, 1, -1, 0, -2, -1, 1, 1, 2, 0, -2, 2, 0, 1, 0, -1, 2, -1, 0, -2, -2, 0, 1, 2, 0, -2, 0, 0, 0, -1, 0, 0, 2, 1, -2, -2, 0, 1, 2, 0, 0, 2, -1, 2, -2, 0, 3, -1, 0, -2, -2, 0, -2, 2, 0, 0, 2, 0, 0, 0, -2, 1, 2, 0, 0, 0, 0, -2, 0, -1, 0, 2, 0, 2, -4, 0, 0, -1, 1, -2, 0, 0, 0, 0, 0, -2, 2, 1, 4, -2, 0, 2, -2, 0, 0, -3, 2, 1, 0, 0, -2, 2, 0

Offset

1,7

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

a(n) is multiplicative with a(2^e) = a(5^e) = (-1)^e, a(p^e) = (1 + (-1)^e)/2 if p == 3, 17, 21, 27, 29, 31, 33, 39 (mod 40), a(p^e) = e+1 if p == 1, 9, 11, 19 (mod 40), a(p^e) = (-1)^e * (e+1) if p == 7, 13, 23, 37 (mod 40).

|a(n)| = A035180(n).

Example

G.f. = q - q^2 + q^4 - q^5 - 2*q^7 - q^8 + q^9 + q^10 + 2*q^11 - 2*q^13 + ...

Mathematica

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; phi[x_] := f[x, x]; A163819[n_] := SeriesCoefficient[ (phi[q]*phi[q^10] - phi[q^2]*phi[q^5])/2, {q, 0, n}]; Table[A163819[n], {n, 1, 50}] (* G. C. Greubel, Aug 05 2017 *)

Prog

(PARI) {a(n) = if( n<1, 0, (qfrep([1, 0; 0, 10], n)[n] - qfrep([2, 0; 0, 5], n)[n]))};
(PARI) {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-10, d)) * if( n%5, kronecker(5, n), (-1)^(0 != sum(k=0, sqrtint(n \ 50), issquare( n/5 - 10*k^2 )))))};

Crossrefs

Cf. A035180.

Sequence in context: A143262 A379590 A035180 * A301734 A281185 A260683

Adjacent sequences: A163816 A163817 A163818 * A163820 A163821 A163822

Keyword

sign,mult

Author

Michael Somos, Aug 04 2009

Status

approved