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A007757
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Dwork-Kontsevich sequence evaluated at 2n.
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3
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1, 2, 36, 144, 1440, 17280, 241920, 2903040, 1567641600, 156764160000, 217275125760000, 1738201006080000, 45193226158080000, 3796230997278720000, 113886929918361600000, 1822190878693785600000, 22489479824838701875200000, 28336744579296764362752000000, 1076796294013277045784576000000, 1679802218660712191423938560000000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
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REFERENCES
| Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps preprint, arXiv:0709.1432
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PROG
| (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 */
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CROSSREFS
| Cf. A131657, A131658, A056612.
Sequence in context: A134785 A143745 A199944 * A141217 A206688 A025531
Adjacent sequences: A007754 A007755 A007756 * A007758 A007759 A007760
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KEYWORD
| nonn
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AUTHOR
| Richard E. Borcherds (reb(AT)math.berkeley.edu)
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EXTENSIONS
| 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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