A123900 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.

6, 12, 60, 180, 2520, 1008, 18144, 18144, 3991680, 5987520, 155675520, 1089728640, 26153487360, 523069747200, 17784371404800, 12312257126400, 935731541606400, 4678657708032, 12772735542927360, 140500090972200960

Offset: 0

Author

Jonathan Sondow, Oct 18 2006

Keywords

• easy
• nonn

Examples

			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.
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

(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}])

Formula

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