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A377221
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.
2
1, -30, 10980, -5822040, 3623245710, -2467207358280, 1779938570782440, -1336872265001920320, 1034337566576031632100, -818707881037376263396710, 659829780447854309255690280, -539628866179308154664183513160, 446708428717281359928910138018680, -373580804664955058627213489276760840
OFFSET
0,2
COMMENTS
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.
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.
LINKS
N. Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains, and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
FORMULA
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.
EXAMPLE
Let F(x) = 1/(E_4(x))^(1/8) = 1 - 30*x + 3780*x^2 - 616440*x^3 + 111056910*x^4 - 21135698280*x^5 + ...
Then
I(F(x)) = 1 - 30*x + 4680*x^2 - 983640*x^3 + 234828510*x^4 - 60324330780*x^5 + ...
I^2(F(x)) = 1 - 30*x + 5580*x^2 - 1431840*x^3 + 422752110*x^4 - 135277163280*x^5 + ...
I^3(F(x)) = 1 - 30*x + 6480*x^2 - 1961040*x^3 + 687787710*x^4 - 262396695780*x^5 + ...
I^4(F(x)) = 1 - 30*x + 7380*x^2 - 2571240*x^3 + 1042895310*x^4 - 461122928280*x^5 + ...
I^5(F(x)) = 1 - 30*x + 8280*x^2 - 3262440*x^3 + 1501034910*x^4 - 753933360780*x^5 + ...
I^6(F(x)) = 1 - 30*x + 9180*x^2 - 4034640*x^3 + 2075166510*x^4 - 1166342993280*x^5 + ...
I^7(F(x)) = 1 - 30*x + 10080*x^2 - 4887840*x^3 + 2778250110*x^4 - 1726904325780*x^5 + ...
I^8(F(x)) = 1 - 30*x + 10980*x^2 - 5822040*x^3 + 3623245710*x^4 - 2467207358280*x^5 + ... = the g.f. A(x).
MAPLE
with(numtheory):
Order := 30:
E_4 := 1 + 240*add(sigma[3](n)*x^n, n = 1..30):
solve(series(x*E_4, x) = y, x):
seq(coeftayl(series((%/y)^(1/8), y), y = 0, n), n = 0..20);
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Peter Bala, Nov 07 2024
STATUS
approved