A123900 - OEIS

%I #12 Jun 04 2019 09:32:00

%S 6,12,60,180,2520,1008,18144,18144,3991680,5987520,155675520,

%T 1089728640,26153487360,523069747200,17784371404800,12312257126400,

%U 935731541606400,4678657708032,12772735542927360,140500090972200960

%N a(n) = (n+3)!/(d(n)*d(n+1)*d(n+2)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

%H J. Sondow, <a href="https://www.jstor.org/stable/27642006">A geometric proof that e is irrational and a new measure of its irrationality</a>, Amer. Math. Monthly 113 (2006) 637-641.

%H J. Sondow, <a href="https://arxiv.org/abs/0704.1282">A geometric proof that e is irrational and a new measure of its irrationality</a>, arXiv:0704.1282 [math.HO], 2007-2010.

%H J. Sondow and K. Schalm, <a href="http://arxiv.org/abs/0709.0671">Which partial sums of the Taylor series for e are convergents to e? (and a link to the primes 2, 5, 13, 37, 463), II</a>, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

%F a(n) = (n+3)!/(A093101(n)*A093101(n+1)*A093101(n+2)) where A093101(n) = gcd(n!,1+n+n(n-1)+...+n!).

%e a(2) = 60 because (2+3)!/(d(2)*d(3)*d(4)) = 5!/(GCD(2,5)*GCD(6,16)*GCD(24,65)) = 120/2 = 60.

%t (A[n_] := If[n==0,1,n*A[n-1]+1]; d[n_] := GCD[A[n],n! ]; Table[(n+3)!/(d[n]*d[n+1]*d[n+2]), {n,0,21}])

%Y Cf. A000522, A061354, A093101, A123899, A123901.

%K easy,nonn

%O 0,1

%A _Jonathan Sondow_, Oct 18 2006