A102473 -id:A102473 - cp's OEIS frontend

A247365 Central terms of triangles A102472 and A102473.

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

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

Original entry on oeis.org

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

A058294 Successive rows of a triangle, the columns of which are generalized Fibonacci sequences S(j).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 2, 1, 1, 3, 7, 10, 7, 3, 1, 1, 4, 13, 30, 43, 30, 13, 4, 1, 1, 5, 21, 68, 157, 225, 157, 68, 21, 5, 1, 1, 6, 31, 130, 421, 972, 1393, 972, 421, 130, 31, 6, 1, 1, 7, 43, 222, 931, 3015, 6961, 9976, 6961, 3015, 931, 222, 43, 7, 1

Offset: 1

Russell Walsmith, Dec 07 2000

From Reinhard Zumkeller, Sep 14 2014: (Start)
T(n,k) = A102473(n,k), k=1..n;
T(n,k) = A102472(n,k-n+1), k=n..2*n-1;
T(n,n) = A001040(n). (End)

			Triangle begins:
  1;
  1, 1, 1;
  1, 2, 3,  2, 1;
  1, 3, 7, 10, 7, 3, 1;
  ...

Reinhard Zumkeller, >Rows n = 1..125 of table, flattened
Russell Walsmith, CL-Chemy Transforms Fibonacci-Type Sequences to Arrays

A001040, A001053, A058307, A058308, A058309 are columns of this triangle.

Cf. A102472, A102473.

a058294 n k = a058294_tabf !! (n-1) !! (k-1)
a058294_row n = a058294_tabf !! (n-1)
a058294_tabf = [1] : zipWith (++) xss (map (tail . reverse) xss)
               where xss = tail a102473_tabl
-- Reinhard Zumkeller, Sep 14 2014

t[n_, n_] = 1; t[n_, k_] := t[n, k] = If[nJean-François Alcover, Oct 05 2016 *)

The j-th column S(j) is generated by a(n+1) = (n+j)*a(n) + a(n-1), a(0)=0, a(1)=1.

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

cp's OEIS Frontend

A247365 Central terms of triangles A102472 and A102473.

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

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A058294 Successive rows of a triangle, the columns of which are generalized Fibonacci sequences S(j).

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Haskell

Mathematica

Formula

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.

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