A001040 a(n+1) = n*a(n) + a(n-1) with a(0)=0, a(1)=1. (Formerly M2863 N1151)

0, 1, 1, 3, 10, 43, 225, 1393, 9976, 81201, 740785, 7489051, 83120346, 1004933203, 13147251985, 185066460993, 2789144166880, 44811373131073, 764582487395121, 13807296146243251, 263103209266016890, 5275871481466581051, 111056404320064218961, 2448516766522879398193

Offset

0,4

Comments

If the initial 0 and 1 are omitted, CONTINUANT transform of 1, 2, 3, 4, 5, ...

a(n+1) is the numerator of the continued fraction given by C(n) = [n, n-1,...,3,2,1], e.g., [1] = 1, [2,1]=3, [3,2,1] = 10/3, [4,3,2,1] = 43/10 etc. Cf. A001053. - Amarnath Murthy, May 02 2001

Along those lines, a(n) is the denominator of the continued fraction [n,n-1,...3,2,1] and is the numerator of the continued fraction [1,2,3,...,n-1]. - Greg Dresden, Feb 20 2020

Starting (1, 3, 10, 43, ...) = eigensequence of triangle A127701. - Gary W. Adamson, Dec 29 2008

For n >=2, a(n) equals the permanent of the (n-1) X (n-1) tridiagonal matrix with 1's along the superdiagonal and the subdiagonal, and consecutive integers from 1 to n along the main diagonal (see Mathematica program below). - John M. Campbell, Jul 08 2011

Generally, solution of the recurrence a(n+1) = n*a(n) + a(n-1) is a(n) = BesselI(n,-2)*(2*a(0)*BesselK(1,2)-2*a(1)*BesselK(0,2)) + (2*a(0)*BesselI(1,2)+2*a(1)*BesselI(0,2))*BesselK(n,2), and asymptotic is a(n) ~ (a(0)*BesselI(1,2)+a(1)*BesselI(0,2)) * (n-1)!. - Vaclav Kotesovec, Jan 05 2013

For n > 0: a(n) = A058294(n,n) = A102473(n,n) = A102472(n,1). - Reinhard Zumkeller, Sep 14 2014

2*n!*a(n) is the number of open tours by a rook on an (n X 2) chessboard which ends at the opposite line of length n. - Mikhail Kurkov, Nov 19 2019

References

Archimedeans Problems Drive, Eureka, 22 (1959), 15.

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

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)

C. Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773. doi: 10.4236/am.2013.45105. See Table 1.

T. Doslic and R. Sharafdini, Hosoya index of splices, bridges and necklaces, Research Gate, 2015.

Tomislav Doslic and R. Sharafdini, Hosoya Index of Splices, Bridges, and Necklaces, in Distance, Symmetry, and Topology in Carbon Nanomaterials, 2016, pp 147-156. Part of the Carbon Materials: Chemistry and Physics book series (CMCP, volume 9), doi:10.1007/978-3-319-31584-3_10.

R. K. Guy, Letters to N. J. A. Sloane, June-August 1968

S. Janson, A divergent generating function that can be summed and analysed analytically, Discrete Mathematics and Theoretical Computer Science; 2010, Vol. 12, No. 2, 1-22.

N. J. A. Sloane, Transforms

Eric Weisstein's World of Mathematics, Continued Fraction Constants

Eric Weisstein's World of Mathematics, Generalized Continued Fraction

Formula

Generalized Fibonacci sequence for (unsigned) Laguerre triangle A021009. a(n+1) = sum{k=0..floor(n/2), C(n-k, k)(n-k)!/k!}. - Paul Barry, May 10 2004

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

E.g.f.: -I*Pi*(BesselY(1, 2*I)*BesselI(0, 2*sqrt(1-x)) - I*BesselI(1, 2)*BesselY(0, 2*I*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

Limit_{n->infinity} a(n)/(n-1)! = BesselI(0,2) = 2.279585302336... (see A070910). - Vaclav Kotesovec, Jan 05 2013

a(n) = 2*(BesselI(0,2)*BesselK(n,2) - BesselI(n,-2)*BesselK(0,2)). - Vaclav Kotesovec, Jan 05 2013

a(n) = (n-1)!*hypergeometric([1-n/2,1/2-n/2],[1,1-n,1-n], 4) for n >= 2. - Peter Luschny, Sep 10 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, Sep 13 2014

Observed: a(n) = A070910*(n-1)!*(1 + 1/(n-1) + 1/(2*(n-1)^2) + O((n-1)^-3)). - A.H.M. Smeets, Aug 19 2018

a(n) mod 2 = A166486(n). - Alois P. Heinz, Jul 03 2023

Example

G.f. = x + x^2 + 3*x^3 + 10*x^4 + 43*x^5 + 225*x^6 + 1393*x^7 + 9976*x^8 + ...

Maple

A001040 := proc(n)
    if n <= 1 then
        n;
    else
        (n-1)*procname(n-1)+procname(n-2) ;
    end if;
end proc: # R. J. Mathar, Mar 13 2015

Mathematica

Table[Permanent[Array[KroneckerDelta[#1, #2]*(#1) + KroneckerDelta[#1, #2 - 1] + KroneckerDelta[#1, #2 + 1] &, {n - 1, n - 1}]], {n, 2, 30}] (* John M. Campbell, Jul 08 2011 *)
Join[{0}, RecurrenceTable[{a[0]==1, a[1]==1, a[n]==n a[n-1]+a[n-2]}, a[n], {n, 30}]] (* Harvey P. Dale, Aug 14 2011 *)
FullSimplify[Table[2(-BesselI[n, -2]BesselK[0, 2]+BesselI[0, 2]BesselK[n, 2]), {n, 0, 20}]] (* Vaclav Kotesovec, Jan 05 2013 *)

Prog

(PARI) {a(n) = contfracpnqn( vector(abs(n), i, i))[1, 2]}; /* Michael Somos, Sep 25 2005 */
(Haskell)
a001040 n = a001040_list !! n
a001040_list = 0 : 1 : zipWith (+)
   a001040_list (zipWith (*) [1..] $ tail a001040_list)
-- Reinhard Zumkeller, Mar 05 2013
(Sage)
def A001040(n):
    if n < 2: return n
    return factorial(n-1)*hypergeometric([1-n/2, -n/2+1/2], [1, 1-n, 1-n], 4)
[round(A001040(n).n(100)) for n in (0..23)] # Peter Luschny, Sep 10 2014
(Magma) a:=[1, 1]; [0] cat [n le 2 select a[n] else (n-1)*Self(n-1) + Self(n-2): n in [1..23]]; // Marius A. Burtea, Nov 19 2019

Crossrefs

A column of A058294. Cf. A001053.

Cf. A127701. - Gary W. Adamson, Dec 29 2008

Similar recurrences: A001053, A058279, A058307. - Wolfdieter Lang, May 19 2010

Cf. A102472, A102473, A166486.

Sequence in context: A030971 A248687 A006932 * A181949 A162286 A032269

Adjacent sequences: A001037 A001038 A001039 * A001041 A001042 A001043

Keyword

easy,nonn,nice,frac

Author

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

Extensions

Definition clarified by A.H.M. Smeets, Aug 19 2018

Status

approved