VOOZH about

URL: https://oeis.org/A246654

⇱ A246654 - OEIS

login
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.
1
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
0,7
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).
EXAMPLE
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
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 *)
PROG
(SageMath)
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)]
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).
Sequence in context: A389492 A357079 A259790 * A370506 A184182 A085771
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 12 2014
STATUS
approved