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A377223 Coefficients of the series whose 12th power is 1/x * series_reversion(x*E_6(x)), where E_6(x) is the Eisenstein series of weight 6.

%I A377223 #15 Nov 17 2024 07:22:18
%S A377223 1,42,34020,39770808,54603156174,82058923220904,130685055490645992,
%T A377223 216707827984305135744,370213729923354622242084,
%U A377223 647073665508052293475274898,1151627718366568095339000345192,2079918757332503030219456972007720,3802403760868562402170776739039126584,7022808067106759130277006634854345528104
%N A377223 Coefficients of the series whose 12th power is 1/x * series_reversion(x*E_6(x)), where E_6(x) is the Eisenstein series of weight 6.
%C A377223 Let R = 1 + x*Z[[x]] denote the set of integral power series with constant term equal to 1. Let P_n = {g^n, g in R}. If f belongs to P_n then the power series 1/x * series_reversion(x*f(x)) is also in P_n. Apply Bala, Theorem 1, Corollary 2.
%C A377223 Here we take f to be the Eisenstein series E_6. See A013973. It is known that the 12th root f^(1/12) has integer coefficients (Heninger et al.). See A109817. It follows that the present sequence is integral.
%H A377223 Peter Bala, <a href="/A251592/a251592.pdf">Fractional iteration of a series inversion operator</a>
%H A377223 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 A377223 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 A377223 G.f.: A(x) = the 12-fold iterate I^12( 1/E_6(x)^(1/12) ), where the operator I : R -> R is defined by I(f(x)) = 1/x * series_reversion(x/f(x)), showing that the g.f. A(x) is integral.
%p A377223 with(numtheory):
%p A377223 Order := 30:
%p A377223 E_6 := 1 - 504*add(sigma[5](n)*x^n, n = 1..30):
%p A377223 solve(series(x*E_6, x) = y, x):
%p A377223 seq(coeftayl(series((%/y)^(1/12), y), y = 0, n), n = 0..20);
%Y A377223 Cf. A013973, A109817, A377221, A377222.
%K A377223 nonn,easy
%O A377223 0,2
%A A377223 _Peter Bala_, Nov 08 2024