OFFSET
0,2
LINKS
Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
FORMULA
a(n) = floor(e^(1/5)*5^n*n!).
From Philippe Deléham, Mar 14 2004: (Start)
a(n) = n!*Sum_{k=0..n} 5^(n-k)/k!.
E.g.f.: exp(x)/(1 - 5*x). (End)
a(n) = Sum_{k=0..n} P(n, k)*5^k. - Ross La Haye, Aug 29 2005
a(n) = hypergeometric_U(1, n+2 , 1/5)/5. - Peter Luschny, Nov 26 2014
From Peter Bala, Mar 01 2017: (Start)
a(n) = Integral_{x >= 0} (5*x + 1)^n*exp(-x) dx.
The e.g.f. y = exp(x)/(1 - 5*x) satisfies the differential equation (1 - 5*x)*y' = (6 - 5*x)*y.
a(n) = (5*n + 1)*a(n-1) - 5*(n - 1)*a(n-2).
The sequence b(n) := 5^n*n! also satisfies the same recurrence with b(0) = 1, b(1) = 5. This leads to the continued fraction representation a(n) = 5^n*n!*( 1 + 1/(5 - 5/(11 - 10/(16 - ... - (5*n - 5)/(5*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/5) = 1 + 1/(5 - 5/(11 - 10/(16 - ... - (5*n - 5)/((5*n + 1) - ... )))). Cf. A010844. (End)
EXAMPLE
a(2) = 5*2*a(1) + 1 = 10*6 + 1 = 61.
MATHEMATICA
m = 16; CoefficientList[E^x/(1-5x) + O[x]^m, x] Range[0, m-1]! (* Jean-François Alcover, Jun 03 2019 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Henry Bottomley, Jun 20 2000
EXTENSIONS
More terms from James A. Sellers, Jul 04 2000
STATUS
approved