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).

1, 1, 1, 2, 2, 36, 36, 144, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 1567641600, 1567641600, 783820800000, 9876142080000, 651825377280000, 217275125760000, 8691005030400000

Offset: 1

Author

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

Keywords

• nonn

Comments

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.

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

Examples

			From _Petros Hadjicostas_, May 24 2020: (Start)
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}.
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)
		

Links

• Christian Krattenthaler, Table of n, a(n) for n = 1..40
• Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
• Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]
• Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151(2) (2010), 175-218.

Crossrefs

Cf. A007757, A056612, A131658.

Formula

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.