A123899 a(n) = (n+1)!/(d(n)*d(n+1)) where d(n) = cancellation factor in reducing Sum_{k=0...n} 1/k! to lowest terms.

1, 2, 3, 12, 60, 360, 252, 2016, 36288, 362880, 4989600, 11975040, 622702080, 8717829120, 65383718400, 5230697472000, 2736057139200, 49249028505600, 30411275102208, 608225502044160, 25545471085854720000

Offset

0,2

Links

Table of n, a(n) for n=0..20.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, Amer. Math. Monthly 113 (2006) 637-641.

J. Sondow, A geometric proof that e is irrational and a new measure of its irrationality, arXiv:0704.1282 [math.HO], 2007-2010.

J. Sondow and K. Schalm, 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, Gems in Experimental Mathematics (T. Amdeberhan, L. A. Medina, and V. H. Moll, eds.), Contemporary Mathematics, vol. 517, Amer. Math. Soc., Providence, RI, 2010.

Formula

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

Example

a(2) = 3 because (2+1)!/(d(2)*d(3)) = 3!/(gcd(2,5)*gcd(6,16)) = 6/2 = 3.

Mathematica

(A[n_] := If[n==0, 1, n*A[n-1]+1]; d[n_] := GCD[A[n], n! ]; Table[(n+1)!/(d[n]*d[n+1]), {n, 0, 22}])

Crossrefs

Cf. A000522, A061354, A093101, A123900, A123901.

Sequence in context: A092980 A191464 A052183 * A188588 A032133 A155579

Adjacent sequences: A123896 A123897 A123898 * A123900 A123901 A123902

Keyword

easy,nonn

Author

Jonathan Sondow, Oct 18 2006

Status

approved