OFFSET
0,2
COMMENTS
Incomplete Gamma Function at 2, more precisely: a(n) = exp(2)*Gamma(1+n,2).
Let P(A) be the power set of an n-element set A. Then a(n) = the total number of ways to add 0 or more elements of A to each element x of P(A) where the elements to add are not elements of x and order of addition is important. - Ross La Haye, Nov 19 2007
a(n) is the number of ways to split the set {1,2,...,n} into two disjoint subsets S,T with S union T = {1,2,...,n} and linearly order S and then choose a subset of T. - Geoffrey Critzer, Mar 10 2009
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.1.2.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..100
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
Ross La Haye, Binary Relations on the Power Set of an n-Element Set, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
FORMULA
a(n) = row sums of A090802. - Ross La Haye, Aug 18 2006
a(n) = n*a(n-1) + 2^n = (n+2)*a(n-1) - (2*n-2)*a(n-2) = n!*Sum_{j=0..n} floor(2^j/j!). - Henry Bottomley, Jul 12 2001
a(n) is the permanent of the n X n matrix with 3's on the diagonal and 1's elsewhere. a(n) = Sum_{k=0..n} A008290(n, k)*3^k. - Philippe Deléham, Dec 12 2003
Binomial transform of A000522. - Ross La Haye, Sep 15 2004
a(n) = Sum_{k=0..n} k!*binomial(n, k)*2^(n-k). - Paul Barry, Apr 22 2005
a(n) = A066534(n) + 2^n. - Ross La Haye, Nov 16 2005
G.f.: hypergeom([1,k],[],x/(1-2*x))/(1-2*x) with k=1,2,3 is the generating function for A010842, A081923, and A082031. - Mark van Hoeij, Nov 08 2011
E.g.f.: 1/E(0), where E(k) = 1 - x/(1-2/(2+(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 21 2011
G.f.: 1/Q(0), where Q(k)= 1 - 2*x - x*(k+1)/(1-x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 18 2013
a(n) ~ n! * exp(2). - Vaclav Kotesovec, Jun 01 2013
From Peter Bala, Sep 25 2013: (Start)
a(n) = n!*e^2 - Sum_{k >= 0} 2^(n + k + 1)/((n + 1)*...*(n + k + 1)).
= n!*e^2 - e^2*( Integral_{t = 0..2} t^n*exp(-t) dt )
= e^2*( Integral_{t >= 2} t^n*exp(-t) dt )
= e^2*( Integral_{t >= 0} t^n*exp(-t)*Heaviside(t-2) dt ),
an integral representation of a(n) as the n-th moment of a nonnegative function on the positive half-axis.
Bottomley's second-order recurrence above a(n) = (n + 2)*a(n-1) - 2*(n - 1)*a(n-2) has n! as a second solution. This yields the finite continued fraction expansion a(n)/n! = 1/(1 - 2/(3 - 2/(4 - 4/(5 - ... - 2*(n - 1)/(n + 2))))) valid for n >= 2. Letting n tend to infinity gives the infinite continued fraction expansion e^2 = 1/(1 - 2/(3 - 2/(4 - 4/(5 - ... - 2*(n - 1)/(n + 2 - ...))))). (End)
a(n) = 2^(n+1)*U(1, n+2, 2), where U is the Bessel U function. - Peter Luschny, Nov 26 2014
For n >= 4, a(n) = r - (r mod 2^(n - floor((2/n) + log_2(n)))) where r = n! * e^2 - 2^(n+1)/(n+1). - Stan Wagon, Apr 28 2016
G.f.: A(x) = 1/(1 - 2*x - x/(1 - x/(1 - 2*x - 2*x/(1 - 2*x/(1 - 2*x - 3*x/(1 - 3*x/(1 - 2*x - 4*x/(1 - 4*x/(1 - 2*x - ... ))))))))). - Peter Bala, May 26 2017
a(n) = Sum_{k=0..n} (-1)^(n-k)*A137346(n, k). - Mélika Tebni, May 10 2022 [This is equivalent to a(n) = KummerU(-n, -n, 2). - Peter Luschny, May 10 2022]
a(n) = F(n), where the function F(x) := 2^(x+1) * Integral_{t >= 0} e^(-2*t)*(1 + t)^x dt smoothly interpolates this sequence to all real values of x. - Peter Bala, Sep 05 2023
MAPLE
G(x):=exp(2*x)/(1-x): f[0]:=G(x): for n from 1 to 19 do f[n]:=diff(f[n-1], x) od: x:=0: seq(f[n], n=0..19); # Zerinvary Lajos, Apr 03 2009
seq(simplify(exp(1)^2*GAMMA(n+1, 2)), n=0..19); # Peter Luschny, Apr 28 2016
seq(simplify(KummerU(-n, -n, 2)), n=0..21); # Peter Luschny, May 10 2022
MATHEMATICA
With[{r = Round[n! E^2 - 2^(n + 1)/(n + 1)]}, r - Mod[r, 2^(n - Floor[2/n + Log2[n]])]] (* for n>=4; Stan Wagon, Apr 28 2016 *)
a[n_] := n! Sum[2^i/i!, {i, 0, n}]
Table[a[n], {n, 0, 21}] (* Gerry Martens , May 06 2016 *)
With[{nn=30}, CoefficientList[Series[Exp[2x]/(1-x), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, May 27 2019 *)
PROG
(PARI) x='x+O('x^44); Vec(serlaplace(exp(2*x)/(1-x))) \\ Joerg Arndt, Apr 29 2016
(Magma) m:=45; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(Exp(2*x)/(1-x))); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Oct 16 2018
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved