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%I M1783 N0704 #118 Apr 26 2024 02:13:41
%S 1,0,1,2,7,30,157,972,6961,56660,516901,5225670,57999271,701216922,
%T 9173819257,129134686520,1946194117057,31268240559432,533506283627401,
%U 9634381345852650,183586751854827751,3681369418442407670,77492344539145388821,1708512949279640961732
%N a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.
%C Denominator of continued fraction given by C(n) = [ 1; 2,3,4,...n ]. Cf. A001040. - _Amarnath Murthy_, May 02 2001
%C If initial 1 is omitted, CONTINUANT transform of 0, 1, 2, 3, 4, 5, ...
%C Number of deco polyominoes of height n having no 1-cell columns. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the vertical and horizontal dominoes are the deco polyominoes of height 2, of which only the vertical domino does not have 1-cell columns. a(n)=A121554(n,0). - _Emeric Deutsch_, Aug 16 2006
%C For positive n, a(n) equals the permanent of the n X n tridiagonal matrix with 1's along the superdiagonal and the subdiagonal, and consecutive integers from 0 to n-1 along the main diagonal (see Mathematica code below). - _John M. Campbell_, Jul 08 2011
%C Conjecture: 2*n!*a(n) is the number of open tours by a rook on an (n X 2) chessboard which starts and ends at the same line of length n. - _Mikhail Kurkov_, Nov 19 2019
%D Archimedeans Problems Drive, Eureka, 20 (1957), 15.
%D M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 35. [From _N. J. A. Sloane_, Jan 29 2009]
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A001053/b001053.txt">Table of n, a(n) for n = 0..100</a>
%H E. Barcucci, A. Del Lungo and R. Pinzani, <a href="http://dx.doi.org/10.1016/0304-3975(95)00199-9">"Deco" polyominoes, permutations and random generation</a>, Theoretical Computer Science, 159, 1996, 29-42.
%H S. B. Ekhad, <a href="http://www.jstor.org/stable/2325130">Problem 10356</a>, Amer. Math. Monthly, 101 (1994), 75. [From _N. J. A. Sloane_, Jan 29 2009]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ContinuedFractionConstants.html">Continued Fraction Constants</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GeneralizedContinuedFraction.html">Generalized Continued Fraction</a>
%F a(n) = a(-n). for all n in Z. - _Michael Somos_, Sep 25 2005
%F E.g.f.: -Pi*(BesselI(1,2)*BesselY(0, 2*I*sqrt(1-x)) + I*BesselY(1, 2*I)*BesselI(0, 2*sqrt(1-x))). Such e.g.f. computations were the result of an e-mail exchange with _Gary Detlefs_. After differentiation and putting x=0 one has to use simplifications. See the Abramowitz-Stegun handbook, p. 360, 9.1.16 and p. 375, 9.63. - _Wolfdieter Lang_, May 19 2010
%F a(n) = 2*K_1(2)*I_n(-2)+2*I_1(2)*K_n(2), where In(z) is the modified Bessel function of the first kind and Kn(x) is the modified Bessel function of the second kind. - _Alexander R. Povolotsky_, Jan 26 2011
%F Limit_{n->infinity} a(n)/(n-1)! = BesselI(1,2) = 1.590636854637329... (A096789). - _Vaclav Kotesovec_, Jan 05 2013, corrected Mar 02 2013
%F a(n+1) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k) * binomial(n-k,k+1). Cf. A058798. - _Peter Bala_, Aug 01 2013
%F a(n) = Gamma(n)*hypergeometric([3/2-n/2, 1-n/2], [2, 2-n, 1-n], 4) for n >= 3. - _Peter Luschny_, Sep 11 2014
%F 0 = a(n)*(-a(n+2)) + a(n+1)*(a(n+1) + a(n+2) - a(n+3)) + a(n+2)*(a(n+2)) for all n in Z. - _Michael Somos_, Feb 09 2017
%F Observed: a(n) = A096789*(n-1)!*(1 + 1/(n-1) + 1/(2*(n-1)^2) + O((n-1)^-3)). - _A.H.M. Smeets_, Aug 19 2018
%e G.f. = 1 + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 157*x^6 + 972*x^7 + 6961*x^8 + ...
%e a(5) = 4*a(4) + a(3) = 4*7+2 = 30.
%e See A058279 and A058307 for similar recurrences and e.g.f.s. - _Wolfdieter Lang_, May 19 2010
%p a[0]:=1: a[1]:=0: for n from 2 to 23 do a[n]:=(n-1)*a[n-1]+a[n-2] od: seq(a[n],n=0..23); # _Emeric Deutsch_, Aug 16 2006
%t a[0]=1; a[1] =0; a[n_]:= (n-1)*a[n-1] + a[n-2]; Table[a[n], {n, 0, 21}] (* _Robert G. Wilson v_, Feb 24 2005 *)
%t a[0] = 1; a[1] = 0; a[n_] := Permanent[SparseArray[{{i_, i_} :> i-1, Band[{2, 1}] -> 1, Band[{1, 2}] -> 1}, {n, n}]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 20}] (* _John M. Campbell_, Jul 08 2011, updated by _Jean-François Alcover_, Nov 14 2016 *)
%t RecurrenceTable[{a[0]==1,a[1]==0,a[n]==(n-1)a[n-1]+a[n-2]},a,{n,30}] (* _Harvey P. Dale_, Jan 31 2013 *)
%t a[ n_] := With[ {m = Abs@n}, If[ m < 2, Boole[m == 0],
%t Gamma[m] HypergeometricPFQ[{3/2 - m/2, 1 - m/2}, {2, 2 - m, 1 - m}, 4]]]; (* _Michael Somos_, Nov 30 2018 *)
%o (PARI) {a(n) = contfracpnqn(vector(abs(n), i, i))[2, 2]}; /* _Michael Somos_, Sep 25 2005 */
%o (Haskell)
%o a001053 n = a001053_list !! n
%o a001053_list = 1 : 0 :
%o zipWith (+) a001053_list (zipWith (*) [1..] $ tail a001053_list)
%o -- _Reinhard Zumkeller_, Nov 02 2011
%o (Sage)
%o def A001053(n):
%o if n < 3: return 1 if n != 1 else 0
%o return gamma(n)*hypergeometric([3/2-n/2,1-n/2], [2,2-n,1-n], 4)
%o [round(A001053(n).n(100)) for n in (0..23)] # _Peter Luschny_, Sep 11 2014
%o (Magma) I:=[0,1]; [1] cat [n le 2 select I[n] else (n-1)*Self(n-1) + Self(n-2): n in [1..25]]; // _G. C. Greubel_, Sep 20 2019
%o (GAP) a:=[0,1];; for n in [3..25] do a[n]:=(n-1)*a[n-1]+a[n-2]; od; Concatenation([1], a); # _G. C. Greubel_, Sep 20 2019
%Y A column of A058294.
%Y The square roots of the terms of A144656.
%Y See also the constant in A060997.
%Y Cf. A121554, A001040. A058798
%K easy,nonn,nice,changed
%O 0,4
%A _N. J. A. Sloane_, _R. K. Guy_
%E More terms from _James A. Sellers_, Sep 19 2000
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