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A010844 a(n) = 2*n*a(n-1) + 1 with a(0) = 1.

%I A010844 #120 Apr 21 2025 13:24:09
%S A010844 1,3,13,79,633,6331,75973,1063623,17017969,306323443,6126468861,
%T A010844 134782314943,3234775558633,84104164524459,2354916606684853,
%U A010844 70647498200545591,2260719942417458913,76864478042193603043,2767121209518969709549,105150605961720848962863
%N A010844 a(n) = 2*n*a(n-1) + 1 with a(0) = 1.
%C A010844 Related to Incomplete Gamma Function at 1/2. - _Michael Somos_, Mar 26 1999
%C A010844 For positive n, a(n) is equal to 2^n times the permanent of the n X n matrix with 3/2's along the main diagonal, and 1's everywhere else. - _John M. Campbell_, Jul 09 2011
%C A010844 Number of ways to sort a spreadsheet with n columns. (A subset of columns is chosen to sort on. These columns are ordered from major to minor, and each designated as to whether to sort by ascending or descending order. For example a spreadsheet with columns A,B,C,D could be sorted by column D ascending, then by column B descending, or any of 632 other ways.) - _Marc LeBrun_, Dec 07 2013
%C A010844 a(n) is a specific instance of sequences having the form b(0) = x, b(n) = a*n*b(n-1) + k for n >= 1. (Here x = 1, a = 2, and k = 1). Sequences of this form have a closed form of b(n) = n!*a^n*x + k*Sum_{j=1..n} n!*a^(n-j)/j!. - _Gary Detlefs_, Mar 26 2018
%D A010844 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 262.
%H A010844 Vincenzo Librandi, <a href="/A010844/b010844.txt">Table of n, a(n) for n = 0..400</a>
%H A010844 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 [alternative scanned copy].
%H A010844 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 A010844 Roland Bacher, <a href="https://doi.org/10.37236/2522">Counting Packings of Generic Subsets in Finite Groups</a>, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
%H A010844 Paul Barry, <a href="https://arxiv.org/abs/1702.04007">Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays</a>, arXiv:1702.04007 [math.CO], 2017.
%H A010844 Mathieu Guay-Paquet and Jeffrey Shallit, <a href="http://dx.doi.org/10.1016/j.disc.2009.06.004">Avoiding Squares and Overlaps Over the Natural Numbers</a>, Discrete Math., 309 (2009), 6245-6254. [From _N. J. A. Sloane_, Nov 27 2009]
%H A010844 Mathieu Guay-Paquet and Jeffrey Shallit, <a href="http://arxiv.org/abs/0901.1397">Avoiding Squares and Overlaps Over the Natural Numbers</a>, arXiv:0901.1397 [math.CO], 2009.
%H A010844 Guo-Niu Han, <a href="https://web.archive.org/web/20111226024706/http://www-irma.u-strasbg.fr:80/~guoniu/papers/p77puzzle.pdf">Enumeration of Standard Puzzles</a> [Wayback Machine link added by _Felix Fröhlich_, Mar 26 2018]
%H A010844 Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>. [Cached copy]
%H A010844 Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H A010844 Michael Z. Spivey and Laura L. Steil, <a href="http://www.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 A010844 a(n) = floor(n! * e^(1/2) * 2^n) = n! * Sum_{k=0..n} 2^(n-k) / k! (i.e. binomial transform of (2n)!! = n!*2^n) = n! * (e^(1/2) * 2^n - Sum_{k >= n+1} 2^(n-k) / k!). - _Michael Somos_, Mar 26 1999
%F A010844 a(n) = A056541(n) + A000165(n). - _Henry Bottomley_, Jun 20 2000
%F A010844 E.g.f.: exp(x)/(1 - 2*x). - _Vladeta Jovovic_, Aug 11 2002
%F A010844 Sum_{n >= 1} 1/a(n) = 0.4246665348160769533082551230... - _Cino Hilliard_, Aug 19 2003
%F A010844 a(n) = Sum_{k=0..n} P(n, k)*2^k, where P(n,k) = n!/(n-k)!. - _Ross La Haye_, Aug 29 2005
%F A010844 G.f.: 1/(1 - x - 2*x/(1 - 2*x/(1 - x - 4*x/(1 - 4*x/(1 - x - 6*x/(1 - 6*x/(1 - x - 8*x/(1 - 8*x/(1 - x - 10*x/(1 - ... (continued fraction).
%F A010844 G.f.: 1/Q(0), where Q(k) = 1 - x*(4*k+3) - 4*x^2*(k+1)^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Sep 30 2013
%F A010844 a(n) = Sum_{k=0..n} C(n,k)*k!*2^k. - _Marc LeBrun_, Dec 07 2013
%F A010844 0 = a(n)*(2*a(n+1) - 5*a(n+2) + a(n+3)) + a(n+1)*(a(n+1) + a(n+2) - a(n+3)) + a(n+2)*a(n+2) if n > -2. - _Michael Somos_, Jan 02 2014
%F A010844 a(n) + (-2*n-1)*a(n-1) + 2*(n-1)*a(n-2) = 0. - _R. J. Mathar_, Jan 31 2014
%F A010844 a(n) = hypergeometric_U(1, n+2, 1/2)/2. - _Peter Luschny_, Nov 26 2014
%F A010844 From _Peter Bala_, Jan 30 2015: (Start)
%F A010844 a(n) = Integral_{x >= 0} (2*x + 1)^n*exp(-x) dx. (Cf. A000354.)
%F A010844 The e.g.f. y = exp(x)/(1 - 2*x) satisfies the differential equation (1 - 2*x)*y' = (3 - 2*x)*y. _R. J. Mathar_'s recurrence above follows easily from this.
%F A010844 The sequence b(n) := 2^n*n! also satisfies _R. J. Mathar_'s recurrence with b(0) = 1 and b(1) = 2. This leads to the continued fraction representation a(n) = 2^n*n!*( 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/(2*n + 1) )))) for n >= 2. Taking the limit gives the continued fraction representation exp(1/2) = 1 + 1/(2 - 2/(5 - 4/(7 - ... - (2*n - 2)/((2*n + 1) - ... )))). (End)
%F A010844 a(n) = 2^n*KummerU(-n, -n, 1/2). - _Peter Luschny_, May 10 2022
%F A010844 a(n) = n!*2^n*hypergeom([-n], [-n], 1/2). - _Peter Luschny_, Jul 28 2024
%e A010844 a(3) = 2*3*a(2) + 1 = 6*13 + 1 = 79.
%e A010844 G.f. = 1 + 3*x + 13*x^2 + 79*x^3 + 633*x^4 + 6331*x^5 + 75973*x^6 + 1063623*x^7 + ...
%p A010844 G:=(x,a,k,n)-> n!*a^n*x + k*sum(n!*a^(n-j)/j!,j=1..n); seq(G(1,2,1,n), n = 0..20) # _Gary Detlefs_, Mar 26 2018
%p A010844 a := n -> 2^n*add((n!/k!)*(1/2)^k, k=0..n):
%p A010844 seq(a(n), n=0..19); # _Peter Luschny_, Jan 06 2020
%p A010844 seq(simplify(2^n*KummerU(-n, -n, 1/2)), n = 0..19); # _Peter Luschny_, May 10 2022
%t A010844 Table[ Gamma[ n, 1/2 ]*Exp[ 1/2 ]*2^(n-1), {n, 1, 24} ]
%t A010844 ...and/or... s=1;lst={};Do[s+=s++n;AppendTo[lst, s], {n, 1, 5!, 2}];lst (* _Vladimir Joseph Stephan Orlovsky_, Oct 23 2008 *)
%t A010844 a[ n_] := If[ n<0, 0, Floor[ n! E^(1/2) 2^n ]] (* _Michael Somos_, Sep 04 2013 *)
%t A010844 nxt[{n_,a_}]:={n+1,2*a(n+1)+1}; NestList[nxt,{0,1},20][[All,2]] (* _Harvey P. Dale_, Jan 06 2022 *)
%t A010844 a[n_] := n! 2^n Hypergeometric1F1[-n, -n, 1/2];
%t A010844 Table[a[n], {n, 0, 19}] (* _Peter Luschny_, Jul 28 2024 *)
%o A010844 (PARI) {a(n) = if( n<0, 0, n! * sum(k=0, n, 2^(n-k) / k!))} /* _Michael Somos_, Sep 04 2013 */
%Y A010844 Cf. A000165, A000354, A000522, A007566, A010845, A019774, A097817, A097819.
%K A010844 easy,nonn
%O A010844 0,2
%A A010844 _Simon Plouffe_
%E A010844 Better description and formulas from _Michael Somos_