A109506 Expansion of (1 - phi(-q)^4)/ 8 in powers of q where phi() is a Ramanujan theta function.

1, -3, 4, -3, 6, -12, 8, -3, 13, -18, 12, -12, 14, -24, 24, -3, 18, -39, 20, -18, 32, -36, 24, -12, 31, -42, 40, -24, 30, -72, 32, -3, 48, -54, 48, -39, 38, -60, 56, -18, 42, -96, 44, -36, 78, -72, 48, -12, 57, -93, 72, -42, 54, -120, 72, -24, 80, -90, 60, -72, 62, -96, 104, -3, 84, -144, 68, -54, 96, -144, 72

Offset: 1

Author

Michael Somos, Jun 30 2005

Keywords

• sign
• easy
• mult

Comments

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

Denoted by xi(n) in Glaisher 1907. - Michael Somos, May 17 2013

Examples

			q - 3*q^2 + 4*q^3 - 3*q^4 + 6*q^5 - 12*q^6 + 8*q^7 - 3*q^8 + 13*q^9 + ...
		

References

• G. Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed, Chelsea Publishing Co., New York 1959 Part II, p. 346 Exercise XXI(18). MR0121327 (22 #12066).
• J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 8).

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000
• J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
• Michael Somos, Introduction to Ramanujan theta functions.
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
• Index entries for sequences mentioned by Glaisher.

Crossrefs

Cf. A000593, A008438, A046897, A096727, A112610.

Cf. A000122, A000700, A010054, A121373.

Programs

Mathematica

a[ n_] := If[ n < 1, 0, -(-1)^n Sum[ If[ Mod[ d, 4] == 0, 0, d], {d, Divisors@n}]] (* Michael Somos, May 17 2013 *)

PARI

{a(n) = if( n<1, 0, -(-1)^n * sumdiv( n, d, if( d%4, d)))}

PARI

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); -1/8 * polcoeff( eta(x + A)^8 / eta(x^2 + A)^4, n))}

Formula

Expansion of (1 - eta(q)^8 / eta(q^2)^4) / 8 in powers of q.

a(n) = Sum_{d divides n} (-1)^(n/d + d) * d [Glaisher].

Multiplicative with a(2^e) = -3, if e>0. a(p^e) = (p^(e+1) - 1) / (p - 1) if p>2.

G.f.: Sum_{k>0} k * (x^k / (1 - x^k) - 6 * x^(2*k) / (1 - x^(2*k)) + 8 * x^(4*k) / (1 - x^(4*k))).

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

a(n) = -(-1)^n * A046897(n). a(n) = -A096727(n) / 8 unless n=0. a(2*n) = -3 * A000593(n). a(2*n + 1) = A008438(n). a(4*n + 1) = A112610(n). a(4*n + 3) = A097723(n).

Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023