A058307 -id:A058307 - cp's OEIS frontend

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

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.

A093858 a(0) = 1, a(1)= 2, a(n) = (a(n+1) - a(n-1))/n, or a(n+1) = n*a(n) + a(n-1).

Original entry on oeis.org

1, 2, 3, 8, 27, 116, 607, 3758, 26913, 219062, 1998471, 20203772, 224239963, 2711083328, 35468323227, 499267608506, 7524482450817, 120890986821578, 2062671258417643, 37248973638339152, 709793170386861531

Offset: 0

Amarnath Murthy, Apr 19 2004

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Similar recurrences: A001040, A001053, A058279, A058307. - Wolfdieter Lang, May 19 2010

a = 1; b = 2; Print[a]; Print[b]; Do[c = n*b + a; Print[c]; a = b; b = c, {n, 1, 30}] (* Ryan Propper, Sep 14 2005 *)
nxt[{n_,a_,b_}]:={n+1,b,b*n+a}; NestList[nxt,{1,1,2},20][[;;,2]] (* Harvey P. Dale, Dec 23 2023 *)

a(n) = -2*(BesselI[n, -2]*(2 BesselK[0, 2] - BesselK[1, 2]) + (-2 BesselI[0, 2] + BesselI[1, -2])*BesselK[n, 2]). - Ryan Propper, Sep 14 2005
E.g.f.: -3*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 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
Lim_{n->infinity} a(n)/(n-1)! = 2*BesselI(0,2) - BesselI(1,-2) = 6.1498074593094635982566633... - Vaclav Kotesovec, Jan 05 2013

a(10)-a(20) from Ryan Propper, Sep 14 2005

A228340 Triangle read by rows: T(n,k) = (n-1)*T(n-1,k) + T(n-2,k), with T(n,n-1)=1, T(n,n-2)=n-2, for n >= 1, 0 <= k <= n-1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 4, 2, 1, 13, 17, 9, 3, 1, 68, 89, 47, 16, 4, 1, 421, 551, 291, 99, 25, 5, 1, 3015, 3946, 2084, 709, 179, 36, 6, 1, 24541, 32119, 16963, 5771, 1457, 293, 49, 7, 1, 223884, 293017, 154751, 52648, 13292, 2673, 447, 64, 8, 1

Offset: 1

N. J. A. Sloane, Aug 29 2013

			Triangle begins:
1,
0,1,
1,1,1,
3,4,2,1,
13,17,9,3,1,
68,89,47,16,4,1,
421,551,291,99,25,5,1,
3015,3946,2084,709,179,36,6,1,
...

Reinhard Zumkeller, Rows n = 1..120 of table, flattened
C. Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.

Diagonals give A058307, A058279, A228341. Row sums give A001040.

a228340 n k = a228340_tabl !! (n-1) !! k
a228340_row n = a228340_tabl !! (n-1)
a228340_tabl = map (reverse . fst) $ iterate f ([1], [1,0]) where
   f (us, vs'@( : vs@(v : ))) = (vs', ws) where
     ws = 1 : (v + 1) : zipWith (+) us (map (* (v + 2)) vs)
-- Reinhard Zumkeller, Aug 31 2013

A228341 Third diagonal (T(n,2)) of triangle in A228340.

Original entry on oeis.org

1, 2, 9, 47, 291, 2084, 16963, 154751, 1564473, 17363954, 209931921, 2746478927, 38660636899, 582656032412, 9361157155491, 159722327675759, 2884363055319153, 54962620378739666, 1102136770630112473, 23199834803611101599

Offset: 2

N. J. A. Sloane, Aug 29 2013

Rick L. Shepherd, Table of n, a(n) for n = 2..200
C. Cannings, The Stationary Distributions of a Class of Markov Chains, Applied Mathematics, Vol. 4 No. 5, 2013, pp. 769-773.

Cf. A228340, A058307, A058279.

Table[FullSimplify[-2*BesselI[1+n,-2] * (BesselK[2,2] + BesselK[3,2]) + 2*(BesselI[2,2] - BesselI[3,2]) * BesselK[1+n,2]],{n,2,20}] (* Vaclav Kotesovec, Feb 14 2014 *)

v = [1, 2]; for(n=4, 21, v = concat(v, n*v[n-2] + v[n-3])); v \\ Rick L. Shepherd, Jan 22 2014

a(2) = 1, a(3) = 2; thereafter, a(n) = n*a(n-1) + a(n-2).

a(n) ~ c * n!, where c = BesselI(2,2)-BesselI(3,2) = 0.47620848845888... - Vaclav Kotesovec, Feb 14 2014

More terms from Rick L. Shepherd, Jan 22 2014

A346960 a(0) = 0, a(1) = 1; a(n) = n * (n+1) * a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 6, 73, 1466, 44053, 1851692, 103738805, 7471045652, 672497847485, 73982234269002, 9766327421355749, 1523621059965765846, 277308799241190739721, 58236371461710021107256, 13977006459609646256481161, 3801803993385285491783983048, 1163365998982356970132155293849

Offset: 0

Ilya Gutkovskiy, Aug 13 2021

a(n) is the numerator of fraction equal to the continued fraction [0; 2, 6, 12, 20, 30, ..., n*(n+1)].

			a(1) =    1 because 1/(1*2)                               = 1/2.
a(2) =    6 because 1/(1*2 + 1/(2*3))                     = 6/13.
a(3) =   73 because 1/(1*2 + 1/(2*3 + 1/(3*4)))           = 73/158.
a(4) = 1466 because 1/(1*2 + 1/(2*3 + 1/(3*4 + 1/(4*5)))) = 1466/3173.

Cf. A001053, A002378, A036245, A058307, A071896, A347051, A347052.

a[0] = 0; a[1] = 1; a[n_] := a[n] = n (n + 1) a[n - 1] + a[n - 2]; Table[a[n], {n, 0, 17}]
Table[Numerator[ContinuedFractionK[1, k (k + 1), {k, 1, n}]], {n, 0, 17}]

a(n) ~ c * n^(2*n + 2) / exp(2*n), where c = 3.2100642122891047165999468271849715691225751316633504931782933233387646256... - Vaclav Kotesovec, Aug 14 2021

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).

cp's OEIS Frontend

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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A093858 a(0) = 1, a(1)= 2, a(n) = (a(n+1) - a(n-1))/n, or a(n+1) = n*a(n) + a(n-1).

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A228340 Triangle read by rows: T(n,k) = (n-1)*T(n-1,k) + T(n-2,k), with T(n,n-1)=1, T(n,n-2)=n-2, for n >= 1, 0 <= k <= n-1.

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A228341 Third diagonal (T(n,2)) of triangle in A228340.

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A346960 a(0) = 0, a(1) = 1; a(n) = n * (n+1) * a(n-1) + a(n-2).

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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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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.