A007757 - OEIS

%I #35 May 24 2020 02:46:03

%S 1,2,36,144,1440,17280,241920,29030400,1567641600,156764160000,

%T 217275125760000,1738201006080000,45193226158080000,

%U 3796230997278720000,113886929918361600000,1822190878693785600000,22489479824838701875200000,28336744579296764362752000000,1076796294013277045784576000000,1679802218660712191423938560000000

%N Dwork-Kontsevich sequence evaluated at 2*n.

%C 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).

%C 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

%H Christian Krattenthaler and Tanguy Rivoal, <a href="http://arxiv.org/abs/0709.1432">On the integrality of the Taylor coefficients of mirror maps</a>, arXiv:0709.1432 [math.NT], 2007-2009.

%H Christian Krattenthaler and Tanguy Rivoal, <a href="https://dx.doi.org/10.4310/CNTP.2009.v3.n3.a5">On the integrality of the Taylor coefficients of mirror maps, II</a>, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]

%H Christian Krattenthaler and Tanguy Rivoal, <a href="https://projecteuclid.org/euclid.dmj/1263478510">On the integrality of the Taylor coefficients of mirror maps</a>, Duke Math. J. 151(2) (2010), 175-218.

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

%t 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]];

%t Array[a, 20] (* _Jean-François Alcover_, Dec 17 2018, from PARI *)

%o (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 */

%Y Cf. A131657, A131658, A056612.

%K nonn

%O 1,2

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

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

%E a(8) corrected by _Sean A. Irvine_, Jan 22 2018