1,2

For n positive, put A_n(z)= sum_j (nj)!/(j!^n) *z^j, B_n(z)= sum_j (nj)!/(j!^n) *z^j * (sum_{j<k<=jn} (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(2n).

A formula, conditional on a widely believed conjecture, can be found in the Krattenthaler-Rivoal paper; see Theorem 4 with k=1 and the remarks on top of page 8. Since Borcherds defined a sequence b(n), but then only entered b(2n) in the Encyclopedia, the formula has to be taken with n replaced by 2n. - Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007

Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps preprint, arXiv:0709.1432

(PARI) {a(n)=local(A, oo=2, c1, c2); if(n<1, 0, n*=2; A=x*O(x^oo); A=exp( sum(j=0, oo, x^j* (n*j)!/(j!^n)* sum(k=j+1, j*n, 1/k), A)/ sum(j=0, oo, 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 */

Cf. A131657, A131658, A056612.

Sequence in context: A145450 A134785 A143745 this_sequence A141217 A025531 A099903

Adjacent sequences: A007754 A007755 A007756 this_sequence A007758 A007759 A007760

nonn

Richard E. Borcherds (reb(AT)math.berkeley.edu)

Definition in comment line, PARI code and terms of sequence corrected by Christian Krattenthaler (christian.krattenthaler(AT)univie.ac.at), Sep 30 2007