A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function.

1, -4, 6, -8, 13, -12, 14, -24, 18, -20, 32, -24, 31, -40, 30, -32, 48, -48, 38, -56, 42, -44, 78, -48, 57, -72, 54, -72, 80, -60, 62, -104, 84, -68, 96, -72, 74, -124, 96, -80, 121, -84, 108, -120, 90, -112, 128, -120, 98, -156, 102, -104, 192, -108, 110

Offset

0,2

Comments

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

Number 33 of the 74 eta-quotients listed in Table I of Martin (1996).

References

J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.179

Hardy, et al., Collected Papers of Srinivasa Ramanujan, p. 326, Question 359.

Links

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

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

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^4) / eta(q^2))^4 in powers of q.

Expansion of q^(-1/2)/4 * k * k' * (K / (Pi/2))^2 in powers of q where k, k', K are Jacobi elliptic functions. - Michael Somos, Jun 22 2012

Euler transform of period 4 sequence [ -4, 0, -4, -4, ...].

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^n, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1 (mod 4), b(p^e) = (-1)^e * (p^(e+1) - 1) / (p - 1) if p == 3 (mod 4).

Given g.f. A(x), then B(x) = 4 * Integral_{0..x} A(x^2) dx = arcsin(4 * x * A001938(x^2)) satisfies 0 = f(B(x), B(x^3)) where f(u, v) = sin(u + v) / 2 - sin((u - v) / 2). - Michael Somos, Oct 14 2013

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = (t/i)^2 f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 27 2013

G.f.: (Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)))^4.

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

G.f.: (Product_{k>0} (1 - x^(2*k - 1)) * (1 - x^(4*k)))^4.

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

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

a(n) = (-1)^n * A008438(n). a(2*n) = A112610(n). a(2*n + 1) = -4 * A097723(n).

Convolution square of A134343. - Michael Somos, Jun 20 2012

a(3*n + 2) = 6 * A258831(n). a(4*n + 3) = -8 * A258835(n). - Michael Somos, Jun 11 2015

Example

G.f. = 1 - 4*x + 6*x^2 - 8*x^3 + 13*x^4 - 12*x^5 + 14*x^6 - 24*x^7 + ...
G.f. = q - 4*q^3 + 6*q^5 - 8*q^7 + 13*q^9 - 12*q^11 + 14*q^13 - 24*q^15 + ...

Mathematica

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ Sqrt[(1 - m) m ] (EllipticK[m] 2/Pi)^2 / (4 q^(1/2)), {q, 0, n}]]; (* Michael Somos, Jun 22 2012 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q] QPochhammer[ q^4] / QPochhammer[ q^2])^4, {q, 0, n}]; (* Michael Somos, Oct 14 2013 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSigma[1, 2 n + 1]]; (* Michael Somos, Jun 15 2015 *)

Prog

(PARI) {a(n) = if( n<0, 0, (-1)^n * sigma(2*n + 1))};
(Sage) A = ModularForms( Gamma0(16), 2, prec=110).basis(); A[1] - 4*A[3]; # Michael Somos, Jun 27 2013
(Magma) A := Basis( ModularForms( Gamma0(16), 2), 110); A[2] - 4*A[4]; /* Michael Somos, Jun 10 2015 */

Crossrefs

Cf. A001938, A008438, A097723, A112610, A134343, A258831, A258835.

Sequence in context: A286372 A151760 A273938 * A008438 A141641 A145284

Adjacent sequences: A121610 A121611 A121612 * A121614 A121615 A121616

Keyword

sign

Author

Michael Somos, Aug 10 2006

Status

approved