A030211 Expansion of q^(-1/2) * (eta(q) * eta(q^2))^4 in powers of q.

1, -4, -2, 24, -11, -44, 22, 8, 50, 44, -96, -56, -121, 152, 198, -160, 176, -48, -162, -88, -198, 52, 22, 528, 233, -200, -242, 88, -176, -668, 550, -264, -44, 188, 224, 728, 154, 484, -1056, -656, -311, 236, -100, -792, 714, 528, 640, -88, -478, 484, 1566, -968, 192, -780, -1994, 648, -942

Offset

0,2

Comments

This is Glaisher's P(n). - N. J. A. Sloane, Nov 24 2018

Number 16 of the 74 eta-quotients listed in Table I of Martin (1996).

References

J. W. L. Glaisher, On the representations of a number as a sum of four squares, and on some allied arithmetical functions, Quarterly Journal of Pure and Applied Mathematics, 36 (1905), 305-358. See p. 340.

J. W. L. Glaisher, The arithmetical functions P(m), Q(m), Omega{m). Quart. J. Math, 37 (1906), 36-48.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)

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. 5).

M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.

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

Eric Weisstein's World of Mathematics, Apéry Number.

Index entries for sequences mentioned by Glaisher

Formula

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

Euler transform of period 2 sequence [ -4, -8, ...]. - Michael Somos, Apr 14 2004

Given g.f. A(x), then B(q) = q * A(q^2) satisfies 0 = f(B(q), B(q^2), B(q^3), B(q^6)) where f(u1, u2, u3, u6) = (81*u6*u3 + u1*u2) * (u2*u3 + u1*u6) + 30 * u1*u2*u3*u6 - 256 * u2^2*u6^2 - 5 * u2^2*u3^2 - 5 * u1^2*u6^2 - u1^2*u3^2. - Michael Somos, Mar 08 2006

Given A = A0 + A1 + A2 + A3 is the 4-section, then 0 = 8 * A0*A2 * (A0^2 + A2^2) + (A1^2 - A3^2) * (A0^2 - A2^2). - Michael Somos, Mar 08 2006

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2. - Michael Somos, Mar 08 2006

G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 64 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, May 28 2013

a(n) = (-1)^n * A134461(n). Convolution square of A002171.

G.f.: exp(4*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

Example

G.f. = 1 - 4*x - 2*x^2 + 24*x^3 - 11*x^4 - 44*x^5 + 22*x^6 + 8*x^7 + 50*x^8 + ...
G.f. = q - 4*q^3 - 2*q^5 + 24*q^7 - 11*q^9 - 44*q^11 + 22*q^13 + 8*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^2])^4, {x, 0, n}]; (* Michael Somos, May 28 2013 *)

Prog

(PARI) {a(n) = if( n<0, 0, polcoeff( (eta(x + x * O(x^n)) * eta(x^2 + x * O(x^n)))^4, n))}; /* Michael Somos, Apr 14 2004 */
(PARI) q='q+O('q^99); Vec((eta(q)*eta(q^2))^4) \\ Altug Alkan, Sep 19 2018
(Sage) CuspForms( Gamma0(8), 4, prec=115).0; # Michael Somos, May 28 2013
(Magma) Basis( CuspForms( Gamma0(8), 4), 115) [1]; /* Michael Somos, May 27 2014 */

Crossrefs

Cf. A002171, A134461 (the same except for signs).

Sequence in context: A336598 A336601 A241437 * A134461 A298593 A228474

Adjacent sequences: A030208 A030209 A030210 * A030212 A030213 A030214

Keyword

sign,look

Author

N. J. A. Sloane.

Status

approved