cp's OEIS Frontend

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

%I A030210 #44 Aug 06 2025 17:32:17
%S A030210 1,-4,2,8,-5,-8,6,0,-23,20,32,16,-38,-24,-10,-64,26,92,100,-40,12,
%T A030210 -128,-78,0,25,152,-100,48,-50,40,-108,256,64,-104,-30,-184,266,-400,
%U A030210 -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
%N A030210 Expansion of (eta(q) * eta(q^5))^4 in powers of q.
%C A030210 Conjecture: |a(p)| < 2*p^(3/2) for p prime. - _Michael Somos_, Oct 31 2005
%C A030210 Unique cusp form of weight 4 for congruence group Gamma_1(5). - _Michael Somos_, Aug 11 2011
%C A030210 Number 13 of the 74 eta-quotients listed in Table I of Martin (1996).
%H A030210 Seiichi Manyama, <a href="/A030210/b030210.txt">Table of n, a(n) for n = 1..10000</a>
%H A030210 M. Koike, <a href="http://projecteuclid.org/euclid.nmj/1118787564">On McKay's conjecture</a>, Nagoya Math. J., 95 (1984), 85-89.
%H A030210 Mathieu Lemire and Kenneth S. Williams, <a href="https://people.math.carleton.ca/~williams/papers/pdf/284.pdf">Evaluation of two convolution sums involving the sum of divisors function</a>, Bulletin of the Australian Mathematical Society, Volume 73, Issue 1 February 2006 , pp. 107-115. See function c5 pp. 107-108.
%H A030210 LMFDB, <a href="https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/5/4/">Space of modular forms of level 5 and weight 4</a>
%H A030210 Y. Martin, <a href="https://doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
%H A030210 Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>
%F A030210 Euler transform of period 5 sequence [ -4, -4, -4, -4, -8, ...]. - _Michael Somos_, May 02 2005
%F A030210 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
%F A030210 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)).
%F A030210 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
%F A030210 G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(5*k)))^4.
%F A030210 Convolution square of A030205. - _Michael Somos_, Jun 15 2014
%e A030210 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 + ...
%t A030210 a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^5])^4, {q, 0, n}]; (* _Michael Somos_, Aug 11 2011 *)
%o A030210 (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))}
%o A030210 (Sage) CuspForms( Gamma1(5), 4, prec = 65).0 # _Michael Somos_, Aug 11 2011
%o A030210 (Magma) Basis( CuspForms( Gamma1(5), 4), 65) [1]; /* _Michael Somos_, May 17 2015 */
%Y A030210 Cf. A030205.
%K A030210 sign,mult
%O A030210 1,2
%A A030210 _N. J. A. Sloane_