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

Author

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

Keywords

• nonn

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

Examples

			G.f. = x + 2*x^2 + 36*x^3 + 144*x^4 + 1440*x^5 + 17280*x^6 + 241920*x^7 + ...
		

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.

Crossrefs

Cf. A131657, A131658, A056612.

Programs

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 *)

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 */

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