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 A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum__{k=j+1..j*n} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)*A_n(z))) has integral coefficients. The sequence is u(n). 8

%I #17 May 24 2020 17:37:10

%S 1,1,1,2,2,36,36,144,144,1440,1440,17280,17280,241920,3628800,

%T 29030400,29030400,1567641600,1567641600,156764160000,49380710400000,

%U 217275125760000,1086375628800000,1738201006080000

%N For n >= 1, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum__{k=j+1..j*n} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)*A_n(z))) has integral coefficients. The sequence is u(n).

%C Different from A131657 and A056612.

%H Christian Krattenthaler, <a href="/A131658/b131658.txt">Table of n, a(n) for n = 1..40</a>

%H Christian Krattenthaler and Tanguy Rivoal, <a href="http://arxiv.org/abs/0709.1432">On the integrality of the Taylor coefficients of mirror maps</a>, arXiv:0709.1432 [math.NT], 2007-2009.

%H Christian Krattenthaler and Tanguy Rivoal, <a href="https://dx.doi.org/10.4310/CNTP.2009.v3.n3.a5">On the integrality of the Taylor coefficients of mirror maps, II</a>, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]

%H Christian Krattenthaler and Tanguy Rivoal, <a href="https://projecteuclid.org/euclid.dmj/1263478510">On the integrality of the Taylor coefficients of mirror maps</a>, Duke Math. J. 151(2) (2010), 175-218.

%F A formula, conditional on a widely believed conjecture, can be found in the article by Krattenthaler and Rivoal (2007-2009) cited in the references: see Theorem 4 and the accompanying remarks.

%Y Cf. A007757 (bisection at even integers), A056612, A131657.

%K nonn

%O 1,4

%A Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007, Sep 30 2007

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Last modified May 28 21:07 EDT 2023. Contains 363028 sequences. (Running on oeis4.)