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A001053 a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.
(Formerly M1783 N0704)
38
1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660, 516901, 5225670, 57999271, 701216922, 9173819257, 129134686520, 1946194117057, 31268240559432, 533506283627401, 9634381345852650, 183586751854827751, 3681369418442407670, 77492344539145388821, 1708512949279640961732 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Denominator of continued fraction given by C(n) = [ 1; 2,3,4,...n ]. Cf. A001040. - Amarnath Murthy, May 02 2001

If initial 1 is omitted, CONTINUANT transform of 0, 1, 2, 3, 4, 5, ...

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

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

2*n!*a(n) is the number of open rook's tours on an (n X 2) chessboard which start and end at the same line of length n. - Mikhail Kurkov, Nov 19 2019

REFERENCES

Archimedeans Problems Drive, Eureka, 20 (1957), 15.

M. E. Larsen, Summa Summarum, A. K. Peters, Wellesley, MA, 2007; see p. 35. [From N. J. A. Sloane, Jan 29 2009]

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n = 0..100

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

S. B. Ekhad, Problem 10356, Amer. Math. Monthly, 101 (1994), 75. [From N. J. A. Sloane, Jan 29 2009]

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Continued Fraction Constants

Eric Weisstein's World of Mathematics, Generalized Continued Fraction

FORMULA

a(n) = a(-n). for all n in Z. - Michael Somos, Sep 25 2005

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

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

Lim {n->inf} a(n)/(n-1)! = BesselI(1,2) = 1.590636854637329... (A096789). - Vaclav Kotesovec, Jan 05 2013, corrected Mar 02 2013

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

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

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

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

EXAMPLE

G.f. = 1 + x^2 + 2*x^3 + 7*x^4 + 30*x^5 + 157*x^6 + 972*x^7 + 6961*x^8 + ...

a(5) = 4*a(4) + a(3) = 4*7+2 = 30.

See A058279 and A058307 for similar recurrences and e.g.f.s. - Wolfdieter Lang, May 19 2010

MAPLE

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

MATHEMATICA

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 *)

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 *)

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 *)

a[ n_] := With[ {m = Abs@n}, If[ m < 2, Boole[m == 0],

Gamma[m] HypergeometricPFQ[{3/2 - m/2, 1 - m/2}, {2, 2 - m, 1 - m}, 4]]]; (* Michael Somos, Nov 30 2018 *)

PROG

(PARI) {a(n) = contfracpnqn(vector(abs(n), i, i))[2, 2]}; /* Michael Somos, Sep 25 2005 */

(Haskell)

a001053 n = a001053_list !! n

a001053_list = 1 : 0 :

   zipWith (+) a001053_list (zipWith (*) [1..] $ tail a001053_list)

-- Reinhard Zumkeller, Nov 02 2011

(Sage)

def A001053(n):

    if n < 3: return 1 if n != 1 else 0

    return gamma(n)*hypergeometric([3/2-n/2, 1-n/2], [2, 2-n, 1-n], 4)

[round(A001053(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 11 2014

(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

(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

CROSSREFS

A column of A058294.

The square roots of the terms of A144656.

See also the constant in A060997.

Cf. A121554, A001040. A058798

Sequence in context: A154259 A066114 A088128 * A124552 A020045 A020135

Adjacent sequences:  A001050 A001051 A001052 * A001054 A001055 A001056

KEYWORD

easy,nonn,nice

AUTHOR

N. J. A. Sloane, R. K. Guy

EXTENSIONS

More terms from James A. Sellers, Sep 19 2000

STATUS

approved

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Last modified May 24 19:14 EDT 2020. Contains 334580 sequences. (Running on oeis4.)