A030210 Expansion of (eta(q) * eta(q^5))^4 in powers of q.

1, -4, 2, 8, -5, -8, 6, 0, -23, 20, 32, 16, -38, -24, -10, -64, 26, 92, 100, -40, 12, -128, -78, 0, 25, 152, -100, 48, -50, 40, -108, 256, 64, -104, -30, -184, 266, -400, -76, 0, 22, -48, 442, 256, 115, 312, -514, -128, -307, -100, 52, -304, 2, 400, -160, 0, 200, 200, 500, -80, -518, 432, -138, -512

Offset

1,2

Comments

Conjecture: |a(p)| < 2*p^(3/2) for p prime. - Michael Somos, Oct 31 2005

Unique cusp form of weight 4 for congruence group Gamma_1(5). - Michael Somos, Aug 11 2011

Number 13 of the 74 eta-quotients listed in Table I of Martin (1996).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

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

Mathieu Lemire and Kenneth S. Williams, Evaluation of two convolution sums involving the sum of divisors function, Bulletin of the Australian Mathematical Society, Volume 73, Issue 1 February 2006 , pp. 107-115. See function c5 pp. 107-108.

LMFDB, Newforms of weight 4 on Gamma_0(5)

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

Formula

Euler transform of period 5 sequence [ -4, -4, -4, -4, -8, ...]. - Michael Somos, May 02 2005

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v^3 + 8*u*v*w + 16*u*w^2 + u^2*w. - Michael Somos, May 02 2005

a(n) is multiplicative with a(5^e) = (-5)^e, a(p^e) = a(p) * a(p^(e-1)) - p^3 * a(p^(e-2)).

G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 25 (t/i)^4 f(t) where q = exp(2 Pi i t). - Michael Somos, Aug 11 2011

G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(5*k)))^4.

Convolution square of A030205. - Michael Somos, Jun 15 2014

Example

G.f. = q - 4*q^2 + 2*q^3 + 8*q^4 - 5*q^5 - 8*q^6 + 6*q^7 - 23*q^9 + 20*q^10 + 32*q^11 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^5])^4, {q, 0, n}]; (* Michael Somos, Aug 11 2011 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x^n * O(x); polcoeff( (eta(x + A) * eta(x^5 + A))^4, n))}
(Sage) CuspForms( Gamma1(5), 4, prec = 65).0 # Michael Somos, Aug 11 2011
(Magma) Basis( CuspForms( Gamma1(5), 4), 65) [1]; /* Michael Somos, May 17 2015 */

Crossrefs

Cf. A030205.

Sequence in context: A030181 A021879 A020806 * A367110 A339991 A098798

Adjacent sequences: A030207 A030208 A030209 * A030211 A030212 A030213

Keyword

sign,mult

Author

N. J. A. Sloane.

Status

approved