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.

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

Author

Peter Luschny, Sep 12 2014

Keywords

• nonn
• tabl

Examples

			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
		

Crossrefs

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.

Programs

Maple

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);

Mathematica

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 *)

Sage

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)]

Formula

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).