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A121613 Expansion of psi(-x)^4 in powers of x where psi() is a Ramanujan theta function. 6
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 (list; graph; refs; listen; history; text; internal format)
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
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
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
Sequence in context: A286372 A151760 A273938 * A008438 A141641 A145284
KEYWORD
sign
AUTHOR
Michael Somos, Aug 10 2006
STATUS
approved

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Last modified March 28 09:04 EDT 2024. Contains 371240 sequences. (Running on oeis4.)