A007757 Dwork-Kontsevich sequence evaluated at 2*n.

1, 2, 36, 144, 1440, 17280, 241920, 29030400, 1567641600, 156764160000, 217275125760000, 1738201006080000, 45193226158080000, 3796230997278720000, 113886929918361600000, 1822190878693785600000, 22489479824838701875200000, 28336744579296764362752000000, 1076796294013277045784576000000, 1679802218660712191423938560000000

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

Table of n, a(n) for n=1..20.

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

Cf. A131657, A131658, A056612.

Sequence in context: A199944 A357445 A227927 * A141217 A206688 A226419

Adjacent sequences: A007754 A007755 A007756 * A007758 A007759 A007760

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