A341875 - OEIS

%I #18 Nov 26 2024 17:21:35

%S 1,30,5310,2453220,910100190,409796742600,181276113779460,

%T 84362079365838960,39636500385830239350,18986938020443181757410,

%U 9186944625290601368703000,4491611148118819794144792660,2212757749022582852433835771860,1097546094982154634980848454416920

%N Coefficients of the series whose 24th power equals E_2(x)*E_4(x)/E_6(x), where E_2(x), E_4(x) and E_6(x) are the Eisenstein series A006352, A004009 and A013973.

%C Since E_2(x)*E_4(x)/E_6(x) == 1 - 24*Sum_{k >= 1} (k - 10*k^3 - 21*k^5)*x^k/(1 - x^k) (mod 144), and since the integer k - 10*k^3 - 21*k^5 is always divisible by 6 it follows that E_2(x)*E_4(x)/E_6(x) == 1 (mod 144). It follows from Heninger et al., p. 3, Corollary 2, that the series expansion of (E_2(x)*E_4(x)/E_6(x))^(1/24) = 1 + 30*x + 5310*x^2 + 2453220*x^3 + 910100190*x^4 + ... has integer coefficients.

%C From _Peter Bala_, Nov 16 2024 (Start):

%C Expansion of ( E_2(x)*E_8(x)/E_10(x) )^(1/24), where E_k(x) is the Eisenstein series of weight k.

%C Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Let P(n) = {g^n, g in R}. The Eisenstein series E_2(x) and E_10(x) lie in P(4) while the series E_8(x) lies in P(16) (Heninger et al.).

%C  We claim that the series (E_2(x)*E_8(x))/E_10(x) belongs to P(24).

%C Proof.

%C E_2(x) = 1 - 24*Sum_{n >= 1} sigma_1(n)*x^n.

%C E_8(x) = 1 + 480*Sum_{n >= 1} sigma_7(n)*x^n.

%C E_10(x) = 1  - 264*Sum_{n >= 1} sigma_9(n)*x^n.

%C Hence, E_2(x)*E_8(x)/E_10(x) == 1 + (12^2)*Sum_{n >= 1} (1/6)*(-sigma_1(n) + 20*sigma_7(n) + 11*sigma_9(n))*x^n (mod 12^2) in R. The polynomial (1/6)*(-k + 20*k^7 + 11*k^9) of degree 9 is integer-valued since it takes integer values for 10 consective values of n (e.g., from n = 0 to n = 9).

%C Hence, E_2(x)*E_8(x)/E_10(x) == 1 (mod 12^2) == 1 (mod (2^4)*(3^2)) in R.

%C It follows from Heninger et al., Theorem 1, Corollary 2, that the series E_2(x)*E_8(x)/E_10(x) belongs to P((2^3)*3) = P(24). End Proof. (End)

%H N. Heninger, E. M. Rains and N. J. A. Sloane, <a href="https://doi.org/10.1016/j.jcta.2006.03.018">On the Integrality of n-th Roots of Generating Functions</a>, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_series">Eisenstein series</a>

%F a(n) ~ c * exp(2*Pi*n) / n^(23/24), where c = 0.0431061156115657949750305669836959595841497962033916083447436... - _Vaclav Kotesovec_, Mar 08 2021

%F Equals the series ( E_2(x)*E_8(x)/E_10(x) )^(1/24). - _Peter Bala_, Nov 16 2024

%p E(2,x) := 1 -  24*add(k*x^k/(1-x^k),   k = 1..20):

%p E(4,x) := 1 + 240*add(k^3*x^k/(1-x^k), k = 1..20):

%p E(6,x) := 1 - 504*add(k^5*x^k/(1-x^k), k = 1..20):

%p with(gfun): series((E(2,x)*E(4,x)/E(6,x))^(1/24), x, 20):

%p seriestolist(%);

%Y Cf. A006352 (E_2), A004009 (E_4), A008410 (E_8), A013973, A013974 (E_10). A108091 (E_8)^(1/16), A110150 ((E_10)^(1/4)), A289392 ((E_2)^(1/4)),  A341871 - A341874, A377973, A377974, A377975, A377976, A377977.

%K nonn,easy

%O 0,2

%A _Peter Bala_, Feb 23 2021