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%I #22 Aug 27 2022 17:22:11
%S 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,
%T 1393,972,421,130,31,6,1,0,9976,6961,3015,931,222,43,7,1,0,81201,
%U 56660,24541,7578,1807,350,57,8,1,0,740785,516901,223884,69133,16485,3193,520,73,9,1,0
%N 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.
%F T(n+k,k) = hypergeom([(1-n)/2, 1-n/2], [1-n, 1+k, 1-n-k], 4)* Pochhammer(k+1, n-1).
%F Recurrence: T(n,k) = T(n-2,k)+(n-1)*T(n-1,k), T(n,n)=0, T(n,n-1)=1.
%F T(n,k) = T(n,-k) = T(-n,k) = T(-n,-k).
%e 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.
%e [ 0, 1, -1, 3, -10, 43, -225, 1393, -9976]
%e [ 1, 0, 1, -2, 7, -30, 157, -972, 6961]
%e [ 1, 1, 0, 1, -3, 13, -68, 421, -3015]
%e [ 3, 2, 1, 0, 1, -4, 21, -130, 931]
%e [ 10, 7, 3, 1, 0, 1, -5, 31, -222]
%e [ 43, 30, 13, 4, 1, 0, 1, -6, 43]
%e [ 225, 157, 68, 21, 5, 1, 0, 1, -7]
%e [1393, 972, 421, 130, 31, 6, 1, 0, 1]
%e [9976, 6961, 3015, 931, 222, 43, 7, 1, 0]
%e The diagonals d(n,k) = T(n+k-floor(n/2),k-floor(n/2)) are represented by polynomials described in A246656.
%e n\k: 0 1 2 3 4 p_n(x)
%e -------------------------------------------------------
%e d(0,k): 0, 0, 0, 0, 0, .. 0 A000004
%e d(1,k): 1, 1, 1, 1, 1, .. 1 A000012
%e d(2,k): [0], 1, 2, 3, 4, .. x A001477
%e d(3,k): [1], 3, 7, 13, 21, .. x^2+x+1 A002061
%e d(4,k): [0, 2], 10, 30, 68, .. x^3+x A034262
%e d(5,k): [1, 7], 43, 157, 421, .. x^4+2*x^3+2*x^2+x+1
%p 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);
%p # Recurrence
%p T := proc(n,k) option remember; local m; m := n-1;
%p if k > m or k < 0 then 0 elif k = m then 1 else T(m-1,k) + m*T(m,k) fi end:
%p seq(print(seq(T(n,k), k=0..n)), n=0..8);
%t 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]]]];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 03 2019 *)
%o (Sage)
%o def A246654_col(n, k): # k-th column of the triangle
%o if n < 2: return n
%o return hypergeometric([(1-n)/2, 1-n/2], [1-n, 1+k, 1-n-k], 4) *rising_factorial(k+1,n-1)
%o for k in range(6): [round(A246654_col(n,k).n(100)) for n in (0..10)]
%Y T(n+0,0) = A001040(n).
%Y T(n+1,1) = A001053(n+1).
%Y T(n+2,2) = A058307(n).
%Y T(n+3,3) = A058308(n).
%Y T(n+4,4) = A058309(n).
%Y Cf. A001040, A001053, A001477, A002061, A034262, A058307, A058308, A058309, A246656.
%K nonn,tabl
%O 0,7
%A _Peter Luschny_, Sep 12 2014
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