OFFSET
1,2
COMMENTS
For n positive, 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 b(n) be the largest integer for which exp(B_n(z)/(b(n)*A_n(z))) has integral coefficients. The sequence is b(2*n).
A formula, conditional on a widely believed conjecture, can be found in the Krattenthaler-Rivoal (2007-2009) paper; see Theorem 4 with k = 1 and the remarks on the top of page 8. Since R. E. Borcherds defined a sequence b(n), but then only entered b(2*n) in the OEIS, the formula has to be taken with n replaced by 2*n. - Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007
LINKS
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.
EXAMPLE
G.f. = x + 2*x^2 + 36*x^3 + 144*x^4 + 1440*x^5 + 17280*x^6 + 241920*x^7 + ...
MATHEMATICA
a[n0_] := Module[{A, MM = 2, n = 2n0, c1, c2}, A = Exp[Sum[x^j (n j)!/ (j!^n) Sum[1/k, {k, j+1, j n}], {j, 0, MM}]/Sum[x^j (n j)!/(j!^n), {j, 0, MM}]]; c1 = SeriesCoefficient[A, {x, 0, 1}]; c2 = SeriesCoefficient[A, {x, 0, 2}]; GCD[c1, (c1 + c1^2)/2 - c2]];
Array[a, 20] (* Jean-François Alcover, Dec 17 2018, from PARI *)
PROG
(PARI) {a(n) = my(A, MM=2, c1, c2); if(n<1, 0, n*=2; A = x * O(x^MM); A = exp( sum(j=0, MM, x^j * (n*j)! / (j!^n) * sum(k=j+1, j*n, 1/k), A) / sum(j=0, MM, x^j * (n*j)! / (j!^n), A)); c1 = polcoeff(A, 1); c2 = polcoeff(A, 2); gcd(c1, (c1 + c1^2)/2 - c2))}; /* Michael Somos, Nov 16 2006 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Richard E. Borcherds (reb(AT)math.berkeley.edu)
EXTENSIONS
Definition in comment line, PARI code and terms of sequence corrected by Christian Krattenthaler (christian.krattenthaler(AT)univie.ac.at), Sep 30 2007
a(8) corrected by Sean A. Irvine, Jan 22 2018
STATUS
approved