A058309 -id:A058309 - cp's OEIS frontend

A058798 a(n) = n*a(n-1) - a(n-2) with a(0) = 0, a(1) = 1.

0, 1, 2, 5, 18, 85, 492, 3359, 26380, 234061, 2314230, 25222469, 300355398, 3879397705, 54011212472, 806288789375, 12846609417528, 217586071308601, 3903702674137290, 73952764737299909, 1475151592071860890

Offset: 0

Christian G. Bower, Dec 02 2000

Note that a(n) = (a(n-1) + a(n+1))/(n+1). - T. D. Noe, Oct 12 2012; corrected by Gary Detlefs, Oct 26 2018
a(n) = log_2(A073888(n)) = log_3(A073889(n)).
a(n) equals minus the determinant of M(n+2) where M(n) is the n X n symmetric tridiagonal matrix with entries 1 just above and below its diagonal and diagonal entries 0, 1, 2, .., n-1. Example: M(4)=matrix([[0, 1, 0, 0], [1, 1, 1, 0], [0, 1, 2, 1], [0, 0, 1, 3]]). - Roland Bacher, Jun 19 2001
a(n) = A221913(n,-1), n>=1, is the numerator sequence of the n-th approximation of the continued fraction -(0 + K_{k>=1} (-1/k)) = 1/(1-1/(2-1/(3-1/(4-... The corresponding denominator sequence is A058797(n). - Wolfdieter Lang, Mar 08 2013
The recurrence equation a(n+1) = (A*n + B)*a(n) + C*a(n-1) with the initial conditions a(0) = 0, a(1) = 1 has the solution a(n) = Sum_{k = 0..floor((n-1)/2)} C^k*binomial(n-k-1,k)*( Product_{j = 1..n-2k-1} (k+j)*A + B ). This is the case A = 1, B = 1, C = -1. - Peter Bala, Aug 01 2013

			Continued fraction approximation 1/(1-1/(2-1/(3-1/4))) = 18/7 = a(4)/A058797(4). - _Wolfdieter Lang_, Mar 08 2013

Seiichi Manyama, Table of n, a(n) for n = 0..449
Svante 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.

Column 1 of A007754.
Cf. A073888, A073889, A221913 (alternating row sums).
Other recurrences of this type: A001040, A036242, A036244, A053983, A053984, A053987, A058307, A058308, A058309, A058797, A058799, A075374, A106174, A121323, A121351, A121353, A121354, A222468, A222470.
Similar sequences: A000806, A001053, A007754, A025164, A093986, A159927, A222467, A222469.

a:=[1,2];; for n in [3..25] do a[n]:=n*a[n-1]-a[n-2]; od; Concatenation([0], a); # Muniru A Asiru, Oct 26 2018

[0] cat [n le 2 select n else n*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 22 2016

t = {0, 1}; Do[AppendTo[t, n*t[[-1]] - t[[-2]]], {n, 2, 25}]; t (* T. D. Noe, Oct 12 2012 *)
nxt[{n_,a_,b_}]:={n+1,b,b*(n+1)-a}; Transpose[NestList[nxt,{1,0,1},20]] [[2]] (* Harvey P. Dale, Nov 30 2015 *)

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

def A058798(n):
    if n < 3: return n
    return hypergeometric([1/2-n/2, 1-n/2],[2, 1-n, -n], -4)*factorial(n)
[simplify(A058798(n)) for n in (0..20)] # Peter Luschny, Sep 10 2014

a(n) = Sum_{k = 0..floor((n-1)/2)} (-1)^k*binomial(n-k-1,k)*(n-k)!/(k+1)!. - Peter Bala, Aug 01 2013
a(n) = A058797(n+1) + A058799(n-1). - Henry Bottomley, Feb 28 2001
a(n) = Pi*(BesselY(1, 2)*BesselJ(n+1, 2) - BesselJ(1,2)* BesselY(n+1,2)). See the Abramowitz-Stegun reference given under A103921, p. 361 eq. 9.1.27 (first line with Y, J and z=2) and p. 360, eq. 9.1.16 (Wronskian). - Wolfdieter Lang, Mar 05 2013
Limit_{n->oo} a(n)/n! = BesselJ(1,2) = 0.576724807756873... See a comment on asymptotics under A084950.
a(n) = n!*hypergeometric([1/2-n/2, 1-n/2], [2, 1-n, -n], -4) for n >= 2. - Peter Luschny, Sep 10 2014

New description from Amarnath Murthy, Aug 17 2002

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.

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.

A246654 T(n,k) = 2(K(n,2)I(k,2) - (-1)^(n+k)I(n,2)K(k,2)), where I(n,x) and K(n,x) are Bessel functions; triangle read by rows for 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 10, 7, 3, 1, 0, 43, 30, 13, 4, 1, 0, 225, 157, 68, 21, 5, 1, 0, 1393, 972, 421, 130, 31, 6, 1, 0, 9976, 6961, 3015, 931, 222, 43, 7, 1, 0, 81201, 56660, 24541, 7578, 1807, 350, 57, 8, 1, 0, 740785, 516901, 223884, 69133, 16485, 3193, 520, 73, 9, 1, 0

Offset: 0

Peter Luschny, Sep 12 2014

			T(n, k) as a rectangular matrix (for n >= 0). Only the lower infinite triangle (0 <= k <=n) constitutes the sequence although T(n,k) is defined for all (n,k) in Z^2.
[   0,    1,   -1,   3, -10,  43, -225, 1393, -9976]
[   1,    0,    1,  -2,   7, -30,  157, -972,  6961]
[   1,    1,    0,   1,  -3,  13,  -68,  421, -3015]
[   3,    2,    1,   0,   1,  -4,   21, -130,   931]
[  10,    7,    3,   1,   0,   1,   -5,   31,  -222]
[  43,   30,   13,   4,   1,   0,    1,   -6,    43]
[ 225,  157,   68,  21,   5,   1,    0,    1,    -7]
[1393,  972,  421, 130,  31,   6,    1,    0,     1]
[9976, 6961, 3015, 931, 222,  43,    7,    1,     0]
The diagonals d(n,k) = T(n+k-floor(n/2),k-floor(n/2)) are represented by polynomials described in A246656.
   n\k:    0   1    2     3    4     p_n(x)
-------------------------------------------------------
d(0,k):    0,  0,   0,    0,   0, .. 0                   A000004
d(1,k):    1,  1,   1,    1,   1, .. 1                   A000012
d(2,k):  [0],  1,   2,    3,   4, .. x                   A001477
d(3,k):  [1],  3,   7,   13,  21, .. x^2+x+1             A002061
d(4,k):  [0,  2],  10,   30,  68, .. x^3+x               A034262
d(5,k):  [1,  7],  43,  157, 421, .. x^4+2*x^3+2*x^2+x+1

T(n+0,0) = A001040(n).
T(n+1,1) = A001053(n+1).
T(n+2,2) = A058307(n).
T(n+3,3) = A058308(n).
T(n+4,4) = A058309(n).
Cf. A001040, A001053, A001477, A002061, A034262, A058307, A058308, A058309, A246656.

T := (n, k) -> (BesselK(n,2)*BesselI(k,2) - (-1)^(n+k)*BesselI(n,2) *BesselK(k,2))*2; seq(lprint(seq(round(evalf(T(n,k),99)), k=0..n)), n=0..8);
# Recurrence
T := proc(n,k) option remember; local m; m := n-1;
if  k > m or k < 0 then 0 elif k = m then 1 else T(m-1,k) + m*T(m,k) fi end:
seq(print(seq(T(n,k), k=0..n)), n=0..8);

T[n_, k_] := T[n, k] = With[{m = n - 1}, If[k > m || k < 0, 0, If[k == m, 1, T[m - 1, k] + m*T[m, k]]]];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 03 2019 *)

def A246654_col(n, k): # k-th column of the triangle
    if n < 2: return n
    return hypergeometric([(1-n)/2, 1-n/2], [1-n, 1+k, 1-n-k], 4) *rising_factorial(k+1,n-1)
for k in range(6): [round(A246654_col(n,k).n(100)) for n in (0..10)]

T(n+k,k) = hypergeom([(1-n)/2, 1-n/2], [1-n, 1+k, 1-n-k], 4)* Pochhammer(k+1, n-1).
Recurrence: T(n,k) = T(n-2,k)+(n-1)*T(n-1,k), T(n,n)=0, T(n,n-1)=1.
T(n,k) = T(n,-k) = T(-n,k) = T(-n,-k).

A246662 a(n) = 2(K(n,2)I(4,2) - (-1)^nI(n,2)K(4,2)) where I(n,x) and K(n,x) are Bessel functions.

Original entry on oeis.org

-10, 7, -3, 1, 0, 1, 5, 31, 222, 1807, 16485, 166657, 1849712, 22363201, 292571325, 4118361751, 62067997590, 997206323191, 17014575491837, 307259565176257, 5854946313840720, 117406185841990657, 2471384848995644517, 54487872863746170031, 1255692460715157555230

Offset: 0

Peter Luschny, Sep 12 2014

sign

Cf. A058309.

a := n -> 2*(BesselK(n,2)*BesselI(4,2)-(-1)^n*BesselI(n,2)* BesselK(4,2)); seq(round(evalf(a(n), 99)), n=0..24);

a(n) = a(-n).

a(n+4) = A058309(n).

cp's OEIS Frontend

A058798 a(n) = n*a(n-1) - a(n-2) 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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A062323 Triangle with a(n,n)=1, a(n,k)=(n-1)*a(n-1,k)+a(n-2,k) for n>k.

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A246654 T(n,k) = 2*(K(n,2)*I(k,2) - (-1)^(n+k)*I(n,2)*K(k,2)), where I(n,x) and K(n,x) are Bessel functions; triangle read by rows for 0 <= k <= n.

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Maple

Mathematica

Sage

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A246662 a(n) = 2*(K(n,2)*I(4,2) - (-1)^n*I(n,2)*K(4,2)) where I(n,x) and K(n,x) are Bessel functions.

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Maple

Formula

A246654 T(n,k) = 2(K(n,2)I(k,2) - (-1)^(n+k)I(n,2)K(k,2)), where I(n,x) and K(n,x) are Bessel functions; triangle read by rows for 0 <= k <= n.

A246662 a(n) = 2(K(n,2)I(4,2) - (-1)^nI(n,2)K(4,2)) where I(n,x) and K(n,x) are Bessel functions.