|
%I #112 Sep 07 2023 12:49:53
%S 1,3,10,38,168,872,5296,37200,297856,2681216,26813184,294947072,
%T 3539368960,46011804672,644165281792,9662479259648,154599668219904,
%U 2628194359869440,47307498477912064,898842471080853504,17976849421618118656,377513837853982588928
%N Expansion of e.g.f.: exp(2*x)/(1-x).
%C Incomplete Gamma Function at 2, more precisely: a(n) = exp(2)*Gamma(1+n,2).
%C 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
%C 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
%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
%D R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Example 5.1.2.
%H T. D. Noe, <a href="/A010842/b010842.txt">Table of n, a(n) for n = 0..100</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary Relations on the Power Set of an n-Element Set</a>, Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6.
%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F a(n) = row sums of A090802. - _Ross La Haye_, Aug 18 2006
%F 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
%F 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
%F Binomial transform of A000522. - _Ross La Haye_, Sep 15 2004
%F a(n) = Sum_{k=0..n} k!*binomial(n, k)*2^(n-k). - _Paul Barry_, Apr 22 2005
%F a(n) = A066534(n) + 2^n. - _Ross La Haye_, Nov 16 2005
%F 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
%F 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
%F 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
%F a(n) ~ n! * exp(2). - _Vaclav Kotesovec_, Jun 01 2013
%F From _Peter Bala_, Sep 25 2013: (Start)
%F a(n) = n!*e^2 - Sum_{k >= 0} 2^(n + k + 1)/((n + 1)*...*(n + k + 1)).
%F = n!*e^2 - e^2*( Integral_{t = 0..2} t^n*exp(-t) dt )
%F = e^2*( Integral_{t >= 2} t^n*exp(-t) dt )
%F = e^2*( Integral_{t >= 0} t^n*exp(-t)*Heaviside(t-2) dt ),
%F an integral representation of a(n) as the n-th moment of a nonnegative function on the positive half-axis.
%F 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)
%F a(n) = 2^(n+1)*U(1, n+2, 2), where U is the Bessel U function. - _Peter Luschny_, Nov 26 2014
%F 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
%F 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
%F 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]
%F 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
%p 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
%p seq(simplify(exp(1)^2*GAMMA(n+1, 2)), n=0..19); # _Peter Luschny_, Apr 28 2016
%p seq(simplify(KummerU(-n, -n, 2)), n=0..21); # _Peter Luschny_, May 10 2022
%t 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 *)
%t a[n_] := n! Sum[2^i/i!, {i, 0, n}]
%t Table[a[n], {n, 0, 21}] (* _Gerry Martens_ , May 06 2016 *)
%t With[{nn=30},CoefficientList[Series[Exp[2x]/(1-x),{x,0,nn}],x] Range[ 0,nn]!] (* _Harvey P. Dale_, May 27 2019 *)
%o (PARI) x='x+O('x^44); Vec(serlaplace(exp(2*x)/(1-x))) \\ _Joerg Arndt_, Apr 29 2016
%o (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
%Y Cf. A053484, A053485, A053486, A008290, A137346.
%Y A010843, A000023, A000166, A000142, A000522, A010842, A053486, A053487 have similar properties.
%K nonn,nice,easy
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
| 