Search: id:A007757
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%I A007757
%S A007757 1,2,36,144,1440,17280,241920,2903040,1567641600,156764160000,
%T A007757 217275125760000,1738201006080000,45193226158080000,
%U A007757 3796230997278720000,113886929918361600000,1822190878693785600000,22489479824838701875200000,
               28336744579296764362752000000,1076796294013277045784576000000,1679802218660712191423938560000000
%N A007757 Dwork-Kontsevich sequence evaluated at 2n.
%C A007757 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).
%C A007757 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
%D A007757 Christian Krattenthaler and Tanguy Rivoal, On the integrality of the 
               Taylor coefficients of mirror maps preprint, arXiv:0709.1432
%o A007757 (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 */
%Y A007757 Cf. A131657, A131658, A056612.
%Y A007757 Sequence in context: A145450 A134785 A143745 this_sequence A141217 A025531 
               A099903
%Y A007757 Adjacent sequences: A007754 A007755 A007756 this_sequence A007758 A007759 
               A007760
%K A007757 nonn
%O A007757 1,2
%A A007757 Richard E. Borcherds (reb(AT)math.berkeley.edu)
%E A007757 Definition in comment line, PARI code and terms of sequence corrected 
               by Christian Krattenthaler (christian.krattenthaler(AT)univie.ac.at), 
               Sep 30 2007

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