This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056546 #30 Jul 02 2025 16:02:00
%S A056546 1,6,61,916,18321,458026,13740781,480927336,19237093441,865669204846,
%T A056546 43283460242301,2380590313326556,142835418799593361,
%U A056546 9284302221973568466,649901155538149792621,48742586665361234446576
%N A056546 a(n) = 5*n*a(n-1) + 1 with a(0)=1.
%H A056546 Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F A056546 a(n) = floor(e^(1/5)*5^n*n!).
%F A056546 From _Philippe Deléham_, Mar 14 2004: (Start)
%F A056546 a(n) = n!*Sum_{k=0..n} 5^(n-k)/k!.
%F A056546 E.g.f.: exp(x)/(1 - 5*x). (End)
%F A056546 a(n) = Sum_{k=0..n} P(n, k)*5^k. - _Ross La Haye_, Aug 29 2005
%F A056546 a(n) = hypergeometric_U(1, n+2 , 1/5)/5. - _Peter Luschny_, Nov 26 2014
%F A056546 From _Peter Bala_, Mar 01 2017: (Start)
%F A056546 a(n) = Integral_{x >= 0} (5*x + 1)^n*exp(-x) dx.
%F A056546 The e.g.f. y = exp(x)/(1 - 5*x) satisfies the differential equation (1 - 5*x)*y' = (6 - 5*x)*y.
%F A056546 a(n) = (5*n + 1)*a(n-1) - 5*(n - 1)*a(n-2).
%F A056546 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)
%e A056546 a(2) = 5*2*a(1) + 1 = 10*6 + 1 = 61.
%t A056546 m = 16; CoefficientList[E^x/(1-5x) + O[x]^m, x] Range[0, m-1]! (* _Jean-François Alcover_, Jun 03 2019 *)
%Y A056546 Cf. A000522, A010844, A010845, A056545, A056547 for analogs. A056546/(A000142*A000351) is an increasingly good approximation to 5th root of e.
%K A056546 nonn,easy
%O A056546 0,2
%A A056546 _Henry Bottomley_, Jun 20 2000
%E A056546 More terms from _James Sellers_, Jul 04 2000