A058294 -id:A058294 - cp's OEIS frontend

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

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

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

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
Conjecture: 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

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

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).

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

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.

a001040 n = a001040_list !! n
a001040_list = 0 : 1 : zipWith (+)
   a001040_list (zipWith (*) [1..] $ tail a001040_list)
-- Reinhard Zumkeller, Mar 05 2013

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

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

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 *)

{a(n) = contfracpnqn( vector(abs(n), i, i))[1, 2]}; /* Michael Somos, Sep 25 2005 */

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

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

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

A001053 a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.

Original entry on oeis.org

1, 0, 1, 2, 7, 30, 157, 972, 6961, 56660, 516901, 5225670, 57999271, 701216922, 9173819257, 129134686520, 1946194117057, 31268240559432, 533506283627401, 9634381345852650, 183586751854827751, 3681369418442407670, 77492344539145388821, 1708512949279640961732

Offset: 0

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

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
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

			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

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).

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

A column of A058294.
The square roots of the terms of A144656.
See also the constant in A060997.
Cf. A121554, A001040. A058798

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

a001053 n = a001053_list !! n
a001053_list = 1 : 0 :
   zipWith (+) a001053_list (zipWith (*) [1..] $ tail a001053_list)
-- Reinhard Zumkeller, Nov 02 2011

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

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

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 *)

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

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

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
Limit_{n->infinity} 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

More terms from James Sellers, Sep 19 2000

A058307 a(n) = (n+1)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 3, 13, 68, 421, 3015, 24541, 223884, 2263381, 25121075, 303716281, 3973432728, 55931774473, 842950049823, 13543132571641, 231076203767720, 4172914800390601, 79516457411189139, 1594502063024173381, 33564059780918830140, 740003817243238436461, 17053651856375402868743

Offset: 0

N. J. A. Sloane, Dec 09 2000

nonn

Numerator of convergent to BesselI(0,2)/BesselI(1,2) for which the continued fraction expansion is [1,2,3....,n]. - Benoit Cloitre, Mar 27 2003

Numerator of continued fraction C(n) minus denominator of continued fraction C(n), where C(n) = [ 1; 2,3,4,...n ]. - Melvin Peralta, Jan 17 2017

G. C. Greubel, Table of n, a(n) for n = 0..445 (terms n=0..100 from T. D. Noe)
Olivier Bodini, Antoine Genitrini, Cécile Mailler, and Mehdi Naima, Strict monotonic trees arising from evolutionary processes: combinatorial and probabilistic study, hal-02865198 [math.CO] / [math.PR] / [cs.DS] / [cs.DM], 2020.
C. Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.
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.
Russell Walsmith, Cl-Chemy II

A column of A058294. Except for first term, -1 times row sums of A053495.

Cf. A001053, A060997, A058798.

a:= function(n)
    if n<2 then return n;
    else return (n+1)*a(n-1) + a(n-2);
    fi;
  end;
List([0..30], n-> a(n) ); # G. C. Greubel, Oct 07 2019

[n le 2 select n-1 else Self(n-2)+Self(n-1)*(n): n in [1..30]]; // Vincenzo Librandi, May 06 2013

A058307 := proc(n) option remember; if n <= 1 then n else A058307(n-2)+(n+1)*A058307(n-1); fi; end;
a:= proc(n) option remember;
      if n<2 then n
    else (n+1)*a(n-1) + a(n-2)
      fi;
    end:
seq(a(n), n=0..30); # G. C. Greubel, Oct 07 2019

RecurrenceTable[{a[0]==0, a[1]==1, a[n]==(n+1)*a[n-1]+a[n-2]}, a, {n, 0, 30}] (* Vincenzo Librandi, May 06 2013 *)
Table[FullSimplify[(-BesselI[2+n,-2] * BesselK[2,2] + BesselI[2,2] * BesselK[2+n,2]) / (BesselI[3,2] * BesselK[2,2] + BesselI[2,2] * BesselK[3,2])],{n,0,20}] (* Vaclav Kotesovec, Feb 14 2014 *)
a[n_]:= a[n]= If[n<2, n, (n+1)*a[n-1] +a[n-2]]; Table[a[n], {n,0,30}] (* G. C. Greubel, Oct 07 2019 *)

my(m=30, v=concat([0,1], vector(m-2))); for(n=3, m, v[n]=n*v[n-1] +v[n-2]); v \\ G. C. Greubel, Nov 24 2018

def A058307(n):
    if n < 2: return n
    return factorial(n+1)*hypergeometric([1/2-n/2,1-n/2], [3,-1-n,1-n], 4)/2
[round(A058307(n).n(100)) for n in (0..21)] # Peter Luschny, Sep 10 2014

@CachedFunction
def a(n):
    if (n<2): return n
    else: return (n+1)*a(n-1) + a(n-2)
[a(n) for n in (0..30)]  # G. C. Greubel, Nov 24 2018

From Wouter Meeussen, Feb 02 2001: (Start)
a(2*r+1) = Sum_{j=0..r} (binomial(r+j, r-j)*(r+j)!/(r-j)! - binomial(r + j, r-j-1)*(r+j+1)!/(r-j)!) and
a(2*r) = Sum_{j=0..r} (binomial(r+j+1, r-j)*(r+j+1)!/(r-j)! - binomial(r +j, r-j)*(r+j+1)!/(r-j+1)! + binomial(r+j+1, r-j)*(r+j+1)!/(r-j)!). (End)
E.g.f.: Pi*(BesselI(2, 2)*BesselY(2, 2*I*sqrt(1-x)) - BesselY(2,2*I)*BesselI(2, 2*sqrt(1-x)))/(1-x). Motivated to look into e.g.f.'s for such recurrences by email exchange with Gary Detlefs. One has to use simplifications after differentiation and putting x=0. See Abramowitz-Stegun handbook p. 360, 9.1.16. - Wolfdieter Lang, May 18 2010
Limit n->infinity a(n)/(n+1)! = BesselI(0,2)-BesselI(1,2) = 0.688948447698738204... - Vaclav Kotesovec, Jan 05 2013
a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)* binomial(n-k+1,k+2). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) = (n+1)!*hypergeometric([1/2-n/2,1-n/2],[3,-1-n,1-n],4)/2 for n >= 2. - Peter Luschny, Sep 10 2014
E.g.f.: 2*(I(2,2)*K(2, 2*sqrt(1-x)) - K(2,2)*I(2, 2*sqrt(1-x)))/(1-x), where I(n, x) and K(n, x) are the modified Bessel functions of the second kind. - G. C. Greubel, Oct 07 2019

A058309 a(n) = (n+3)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 5, 31, 222, 1807, 16485, 166657, 1849712, 22363201, 292571325, 4118361751, 62067997590, 997206323191, 17014575491837, 307259565176257, 5854946313840720, 117406185841990657, 2471384848995644517, 54487872863746170031, 1255692460715157555230

Offset: 0

N. J. A. Sloane, Dec 09 2000

nonn

			G.f. = x + 5*x^2 + 31*x^3 + 222*x^4 + 1807*x^5 + 16485*x^6 + 166657*x^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..445
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.
Russell Walsmith, DCL-Chemy II: Reflections and Other Symmetries

A column of A058294. Cf. A058798.

I:=[1,5]; [0] cat [n le 2 select I[n] else (n+3)*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 24 2018

bi:=n->BesselI(4+n,-2); bk:=n->BesselK(4+n,2); i:=n->BesselI(n,2);
k:=n->BesselK(n,2); a := n ->(bi(n)*(5*k(5)-k(6))+bk(n)*(5*i(5) +i(6)))/(i(6)*k(5)+i(5)*k(6)); seq(round(evalf(a(n),99)),n=0..20); # Peter Luschny, Sep 11 2014

a[0] = 0; a[1] = 1; a[n_] := a[n - 2] + (n + 3)*a[n - 1]; Table[a[n], {n, 0, 20}] (* Wesley Ivan Hurt, Sep 12 2014 *)
a[ n_] := With[{m = n + 4}, 2 (BesselK[m, 2] BesselI[4, 2] - (-1)^m BesselI[m, 2] BesselK[4, 2]) // FullSimplify]; (* Michael Somos, Dec 09 2014 *)
a[ n_] := With[{m = Abs[n + 4]}, If[ m < 5, {-10, 7, -3, 1, 0}[[m + 1]], (m - 1)! HypergeometricPFQ[ {5/2 - m/2, 3 - m/2}, {5, 1 - m, 5 - m}, 4] / 24]]; (* Michael Somos, Dec 09 2014 *)
nxt[{n_,a_,b_}]:={n+1,b,b(n+4)+a}; NestList[nxt,{1,0,1},20][[;;,2]] (* Harvey P. Dale, Jul 30 2024 *)

a(n)=round((besseli(n+4,-2)*(5*besselk(5,2)-besselk(6,2)) + besselk(n+4,2)*(5*besseli(5,2) + besseli(6,2))) / (besseli(6,2)*besselk(5,2) + besseli(5,2)*besselk(6,2))) \\ Charles R Greathouse IV, Sep 11 2014

m=30; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=(n+2)*v[n-1] +v[n-2]); v \\ G. C. Greubel, Nov 24 2018

def A058309(n):
    if n < 2: return n
    return factorial(n+3)*hypergeometric([1/2-n/2, 1-n/2], [5, -n-3, 1-n], 4)/24
[round(A058309(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 12 2014

@cached_function
def A058309(n):
    if (n==0): return 0
    elif (n==1): return 1
    else: return (n+3)*A058309(n-1) + A058309(n-2)
[A058309(n) for n in range(30)] # G. C. Greubel, Nov 24 2018

a(n) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)* binomial(n-k+3,k+4). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) = (I(n+4,-2)*(5*K_5-K_6)+K(n+4,2)*(5*I_5+I_6))/(I_6*K_5+I_5* K_6), where I_n and K_n are the Bessel functions of the first respectively the second kind, all evaluated at x=2. - Peter Luschny, Sep 11 2014
a(n) = (n+3)!*hypergeometric([1/2-n/2, 1-n/2], [5, -n-3, 1-n], 4)/24 for n >= 2. - Peter Luschny, Sep 12 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
a(-8-n) = a(n) for all n in Z. - Michael Somos, Sep 13 2014
a(n) ~ c * n! * n^3, where c = 7*BesselI(0,2) - 10*BesselI(1,2) = 0.050728569979180238237886835684070993456106124542846907172998415564687... - Vaclav Kotesovec, May 05 2024

A058308 a(n) = (n+2)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 4, 21, 130, 931, 7578, 69133, 698908, 7757121, 93784360, 1226953801, 17271137574, 260294017411, 4181975416150, 71353876091961, 1288551745071448, 24553837032449473, 492365292394060908, 10364224977307728541, 228505314793164088810

Offset: 0

N. J. A. Sloane, Dec 09 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 0..445
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.
Russell Walsmith, CL-Chemy II: Reflections and Other Symmetries

A column of A058294.
Similar recurrences: A001040, A001053, A058279, A058307, A093858. - Wolfdieter Lang, May 19 2010
Cf. A058798.

I:=[1,4]; [0] cat [n le 2 select I[n] else (n+2)*Self(n-1) +Self(n-2): n in [1..30]]; // G. C. Greubel, Nov 24 2018

RecurrenceTable[{a[0]==0,a[1]==1,a[n]==a[n-2]+(n+2)a[n-1]},a[n], {n,20}] (* Harvey P. Dale, May 21 2011 *)
FullSimplify[Table[(-4*BesselI[3+n,-2]*BesselK[4,2] + BesselI[3+n,-2]*BesselK[5,2] + 4*BesselI[4,2]*BesselK[3+n,2] + BesselI[5,2]*BesselK[3+n,2]) / (BesselI[5,2]*BesselK[4,2] + BesselI[4,2]*BesselK[5,2]),{n,0,20}]] (* Vaclav Kotesovec, Oct 05 2013 *)

m=30; v=concat([1,4], vector(m-2)); for(n=3, m, v[n]=(n+2)*v[n-1] +v[n-2]); concat([0], v) \\ G. C. Greubel, Nov 24 2018

def A058308(n):
    if n < 2: return n
    return factorial(n+2)*hypergeometric([1/2-n/2, 1-n/2], [4, -n-2, 1-n], 4)/6
[round(A058308(n).n(100)) for n in (0..20)] # Peter Luschny, Sep 10 2014

@cached_function
def A058308(n):
    if n==0: return 0
    if n==1: return 1
    return (n+2)*A058308(n-1) + A058308(n-2)
[A058308(n) for n in range(30)]  # G. C. Greubel, Nov 24 2018

E.g.f.: -Pi*(BesselY(4, 2*i)*BesselI(3, 2*sqrt(1-x)) - i*BesselI(4, 2)*BesselY(3, 2*i*sqrt(1-x)))/(1-x)^(3/2). Such e.g.f. computations were the result of an e-mail exchange with Gary Detlefs. After differentiation and setting 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) = Sum_{k = 0..floor((n-1)/2)} (n-2*k-1)!*binomial(n-k-1,k)* binomial(n-k+2,k+3). Cf. A058798. - Peter Bala, Aug 01 2013
a(n) = (-4*BesselI(3+n,-2)*BesselK(4,2) + BesselI(3+n,-2)*BesselK(5,2) + 4*BesselI(4,2)*BesselK(3+n,2) + BesselI(5,2)*BesselK(3+n,2)) / (BesselI(5,2)*BesselK(4,2) + BesselI(4,2)*BesselK(5,2)). - Vaclav Kotesovec, Oct 05 2013
a(n) ~ (4*BesselI(4,2) + BesselI(5,2))/(BesselI(5,2)*BesselK(4,2) + BesselI(4,2)*BesselK(5,2)) * sqrt(Pi/2) * n^(n+5/2)/exp(n). - Vaclav Kotesovec, Oct 05 2013
a(n) = (n+2)!*hypergeometric([1/2-n/2,1-n/2],[4,-n-2,1-n],4)/6 for n >= 2. - Peter Luschny, Sep 10 2014
a(n) = (n+1)! [x^(n+1)] (2/(1-x))*(K(3,2)*I(2,2*sqrt(1-x))+I(3,2)*K(2,2*sqrt(1-x))), K and I Bessel functions. - Peter Luschny, May 01 2017

A102472 Triangle read by rows. Let S(k) be the sequence defined by F(0)=0, F(1)=1, F(n-1) + (n+k)*F(n) = F(n+1). E.g. S(0) = 0, 1, 1, 3, 10, 43, 225, 1393, 9976, 81201, ... Then S(0), S(1), S(2), ... are written vertically, next to each other, with the initial term of each on the next row down.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 7, 3, 1, 43, 30, 13, 4, 1, 225, 157, 68, 21, 5, 1, 1393, 972, 421, 130, 31, 6, 1, 9976, 6961, 3015, 931, 222, 43, 7, 1, 81201, 56660, 24541, 7578, 1807, 350, 57, 8, 1, 740785, 516901, 223884, 69133, 16485, 3193, 520, 73, 9, 1

Offset: 1

Russell Walsmith, Jan 09 2005

T(n,1) = A001040(n); T(n,k) = A058294(n,n+k-1), k = 1..n. - Reinhard Zumkeller, Sep 14 2014

This triangle results when the first column is removed from A062323. - Georg Fischer, Jul 26 2023

			Triangle begins:
[1] 1;
[2] 1, 1;
[3] 3, 2, 1;
[4] 10, 7, 3, 1;
[5] 43, 30, 13, 4, 1;
[6] 225, 157, 68, 21, 5, 1;
[7] 1393, 972, 421, 130, 31, 6, 1;
[8] 9976, 6961, 3015, 931, 222, 43, 7, 1;

Reinhard Zumkeller, Table of n, a(n) for n = 1..7875

Mirror image of triangle in A102473.

Cf. A001040, A058294, A062323, A247365 (central terms).

a102472 n k = a102472_tabl !! (n-1) !! (k-1)
a102472_row n = a102472_tabl !! (n-1)
a102472_tabl = map reverse a102473_tabl
-- Reinhard Zumkeller, Sep 14 2014

Entry revised by N. J. A. Sloane, Jul 09 2005

A102473 Triangle read by rows. Let S(k) be the sequence defined by F(0)=0, F(1)=1, F(n-1) + (n+k)*F(n) = F(n+1). E.g. S(0) = 0,1,1,3,10,43,225,1393,9976,81201, ... Then S(0), S(1), S(2), ... are written next to each other, vertically, with the initial term of each on the next row down. The order of the terms in the rows are then reversed.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 10, 1, 4, 13, 30, 43, 1, 5, 21, 68, 157, 225, 1, 6, 31, 130, 421, 972, 1393, 1, 7, 43, 222, 931, 3015, 6961, 9976, 1, 8, 57, 350, 1807, 7578, 24541, 56660, 81201, 1, 9, 73, 520, 3193, 16485, 69133, 223884, 516901, 740785, 1, 10, 91, 738

Offset: 1

Russell Walsmith (russw(AT)lycos.com), Jan 09 2005

For this triangle, the algorithm that generates the Bernoulli numbers gives 3/2, then 1/6, 1/24, ... 1/n!

T(n,n) = A001040(n); T(n,k) = A058294(n,k), k = 1..n. - Reinhard Zumkeller, Sep 14 2014

			Triangle begins:
0
0 1
0 1 1
0 1 2 3
0 1 3 7 10
0 1 4 13 30 43
...
(the zeros are omitted).

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

Mirror image of triangle in A102472.

Cf. A001040, A058294, A247365 (central terms).

a102473 n k = a102473_tabl !! (n-1) !! (k-1)
a102473_row n = a102473_tabl !! (n-1)
a102473_tabl = [1] : [1, 1] : f [1] [1, 1] 2 where
   f us vs x = ws : f vs ws (x + 1) where
               ws = 1 : zipWith (+) ([0] ++ us) (map (* x) vs)
-- Reinhard Zumkeller, Sep 14 2014

Entry revised by N. J. A. Sloane, Jul 09 2005

A062323 Triangle with a(n,n)=1, a(n,k)=(n-1)*a(n-1,k)+a(n-2,k) for n>k.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 2, 1, 7, 10, 7, 3, 1, 30, 43, 30, 13, 4, 1, 157, 225, 157, 68, 21, 5, 1, 972, 1393, 972, 421, 130, 31, 6, 1, 6961, 9976, 6961, 3015, 931, 222, 43, 7, 1, 56660, 81201, 56660, 24541, 7578, 1807, 350, 57, 8, 1, 516901, 740785, 516901, 223884

Offset: 0

Henry Bottomley, Jul 05 2001

			Triangle starts:
[0] 1;
[1] 0, 1;
[2] 1, 1, 1;
[3] 2, 3, 2, 1;
[4] 7, 10, 7, 3, 1;
[5] 30, 43, 30, 13, 4, 1;
[6] 157, 225, 157, 68, 21, 5, 1;
[7] 972, 1393, 972, 421, 130, 31, 6, 1;
[8] 6961, 9976, 6961, 3015, 931, 222, 43, 7, 1;

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Essentially the same as A058294, but more easy seen as a triangle. Columns include A001040, A001053, A058307, A058308, A058309. Other sequences appearing on the right hand side include A000012, A001477, A002061, A034262.

a062323 n k = a062323_tabl !! n !! k
a062323_row n = a062323_tabl !! n
a062323_tabl = map fst $ iterate f ([1], [0,1]) where
   f (us, vs) = (vs, ws) where
     ws = (zipWith (+) (us ++ [0]) (map (* v) vs)) ++ [1]
          where v = last (init vs) + 1
-- Reinhard Zumkeller, Mar 05 2013

a(n, k)=k*a(n, k+1)+a(n, k+2) for n>k.

A247365 Central terms of triangles A102472 and A102473.

Original entry on oeis.org

1, 2, 13, 130, 1807, 32280, 705421, 18237164, 544505521, 18438430990, 698246022001, 29239344782022, 1341545985079903, 66926098621724300, 3606825675219961657, 208826700420103831480, 12926842112341879416001, 851962999949978920707834, 59561112879709434549509941

Offset: 1

Reinhard Zumkeller, Sep 14 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..300

Cf. A102472, A102473, A058294.

Haskell
```
a247365 n = a102473 (2 * n - 1) n
```

seq(round(2*BesselI(n-1,2)*BesselK(2*n-1,2)), n=1..30); # Mark van Hoeij, Nov 08 2022
A001040 := proc(n) options remember;
  if n < 2 then n else (n - 1)*procname(n-1) + procname(n-2) fi
end:
A001053 := proc(n) options remember;
  if n < 2 then 1-n else (n - 1)*procname(n-1) + procname(n-2) fi
end:
seq( (-1)^n * (A001040(n-1) * A001053(2*n-1) - A001053(n-1) * A001040(2*n-1)), n=1..30); # Mark van Hoeij, Jul 10 2024

Table[DifferenceRoot[Function[{y,m},{y[2+m]==(m+n)y[1+m]+y[m],y[0]==0,y[1]==1}]][n],{n,1,20}] (* Benedict W. J. Irwin, Nov 03 2016 *)

a(n) = A102472(2*n-1,n) = A102473(2*n-1,n).
a(n) = y(n,n), where y(m+2,n) = (m + n)*y(m+1,n) + y(m,n), with y(0,n)=0, y(1,n)=1 for all n. - Benedict W. J. Irwin, Nov 03 2016
a(n) = round(2*BesselI(n-1,2)*BesselK(2*n-1,2)). - Mark van Hoeij, Nov 08 2022
a(n) ~ 2^(2*n - 3/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Nov 09 2022
a(n) = (-1)^n * (A001040(n-1) * A001053(2*n-1) - A001053(n-1) * A001040(2*n-1)). - Mark van Hoeij, Jul 10 2024

cp's OEIS Frontend

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

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A001053 a(n+1) = n*a(n) + a(n-1) with a(0)=1, a(1)=0.

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A058307 a(n) = (n+1)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

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A058309 a(n) = (n+3)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

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A058308 a(n) = (n+2)*a(n-1) + a(n-2), with a(0)=0, a(1)=1.

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