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A131657 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=1..j*n} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)*A_n(z))) has integral coefficients. The sequence is b(n).

%I A131657 #37 Sep 22 2024 07:25:01
%S A131657 1,1,1,2,2,36,36,144,144,1440,1440,17280,17280,241920,3628800,
%T A131657 29030400,29030400,1567641600,1567641600,783820800000,9876142080000,
%U A131657 651825377280000,217275125760000,8691005030400000
%N A131657 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=1..j*n} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)*A_n(z))) has integral coefficients. The sequence is b(n).
%C A131657 Different from A131658 and A056612. The first difference between A056612 and this sequence occurs for n = 20, while the first difference between A056612 and A131658 occurs for n = 21.
%C A131657 Krattenhaler and Rivoal (2007-2009) conjecture that a(n) = n!*Xi(n), where Xi(1) = 1, Xi(7) = 1/140, and Xi(n) = Product_{p <= n} p^min(2, v_p(H_n)) for n <> 1, 7, where v_p(r) is the p-adic valuation of rational r. (Here p indicates a prime and H_n is the n-th harmonic number.) - _Petros Hadjicostas_, May 24 2020
%H A131657 Christian Krattenthaler, <a href="/A131657/b131657.txt">Table of n, a(n) for n = 1..40</a>
%H A131657 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 A131657 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 A131657 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 A131657 A formula, conditional on a widely believed conjecture, can be found in Theorem 3 with k = 1 in the article by Krattenthaler and Rivoal (2007-2009) cited in the references; see the remarks before Theorem 4 in that article.
%e A131657 From _Petros Hadjicostas_, May 24 2020: (Start)
%e A131657 To illustrate the Krattenhaler-Rivoal conjecture consider the case n = 24. Then H_24 = Sum_{k=1..24} 1/k = 1347822955/356948592 and {p <= 24} = {2, 3, 5, 7, 11, 13, 17, 19, 23} with {v_p(numerator): p <= 24} = {0, 0, 1, 0, 0, 0, 0, 0, 0} and {v_p(denominator): p <= 24} = {4, 1, 0, 1, 1, 1, 1, 1, 1}.
%e A131657 Thus, the conjectured value for a(24) is 24! * (2^(0-4) * 3^(0-1) * 5^(1-0) * 7^(0-1) * 11^(0-1) * 13^(0-1) * 17^(0-1) * 19^(0-1) * 23^(0-1)) since no exponent of a prime is > 2. This product equals 8691005030400000 = a(24). (End)
%Y A131657 Cf. A007757, A056612, A131658.
%K A131657 nonn
%O A131657 1,4
%A A131657 Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007, Sep 30 2007