A377221 - OEIS

%I #17 Nov 17 2024 07:21:50

%S 1,-30,10980,-5822040,3623245710,-2467207358280,1779938570782440,

%T -1336872265001920320,1034337566576031632100,

%U -818707881037376263396710,659829780447854309255690280,-539628866179308154664183513160,446708428717281359928910138018680,-373580804664955058627213489276760840

%N Coefficients of the series whose 8th power is 1/x * series_reversion(x * E_4(x)), where E_4(x) is the Eisenstein series of weight 4.

%C Let R = 1 + x*Z[[x]] denote the set of integer power series with constant term equal to 1. Define the operator T: R -> R by T(f(x)) = 1/x * series_reversion(x*f(x)). Let P_n = {g^n, g in R}. It follows from Bala, Theorem 1, Corollary 2, that if f belongs to P_n then T(f) is also in P_n.

%C Here we take f to be the Eisenstein series E_4, the theta series of the E_8 lattice. See A004009. It is known that the 8th root E_4^(1/8) has integer coefficients (Heninger et al.). It follows that the present sequence is integral.

%H Peter Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>

%H N. Heninger, E. M. Rains, and N. J. A. Sloane, <a href="https://arxiv.org/abs/math/0509316">On the Integrality of n-th Roots of Generating Functions</a>, arXiv:math/0509316 [math.NT], 2005-2006.

%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.

%F G.f.: A(x) = the 8-fold iterate I^8( 1/(E_4(x))^(1/8) ), where I : R -> R denotes the operator I(f(x)) = 1/x * series_reversion(x/f(x)), showing that the g.f. A(x) is integral.

%e Let F(x) = 1/(E_4(x))^(1/8) = 1 - 30*x + 3780*x^2 - 616440*x^3 + 111056910*x^4 - 21135698280*x^5 + ...

%e Then

%e I(F(x))   = 1 - 30*x + 4680*x^2 - 983640*x^3 + 234828510*x^4 - 60324330780*x^5 + ...

%e I^2(F(x)) = 1 - 30*x + 5580*x^2 - 1431840*x^3 + 422752110*x^4 - 135277163280*x^5 + ...

%e I^3(F(x)) = 1 - 30*x + 6480*x^2 - 1961040*x^3 + 687787710*x^4 - 262396695780*x^5 + ...

%e I^4(F(x)) = 1 - 30*x + 7380*x^2 - 2571240*x^3 + 1042895310*x^4 - 461122928280*x^5 + ...

%e I^5(F(x)) = 1 - 30*x + 8280*x^2 - 3262440*x^3 + 1501034910*x^4 - 753933360780*x^5 + ...

%e I^6(F(x)) = 1 - 30*x + 9180*x^2 - 4034640*x^3 + 2075166510*x^4 - 1166342993280*x^5 + ...

%e I^7(F(x)) = 1 - 30*x + 10080*x^2 - 4887840*x^3 + 2778250110*x^4 - 1726904325780*x^5 + ...

%e I^8(F(x)) = 1 - 30*x + 10980*x^2 - 5822040*x^3 + 3623245710*x^4 - 2467207358280*x^5 + ... = the g.f. A(x).

%p with(numtheory):

%p Order := 30:

%p E_4 := 1 + 240*add(sigma[3](n)*x^n, n = 1..30):

%p solve(series(x*E_4, x) = y, x):

%p seq(coeftayl(series((%/y)^(1/8), y), y = 0, n), n = 0..20);

%Y Cf. A004009, A108091, A377220, A377223.

%K sign,easy

%O 0,2

%A _Peter Bala_, Nov 07 2024