A268434 Triangle read by rows, Lah numbers of order 2, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k), for n>=0 and 0<=k<=n.
1, 0, 1, 0, 2, 1, 0, 10, 10, 1, 0, 100, 140, 28, 1, 0, 1700, 2900, 840, 60, 1, 0, 44200, 85800, 31460, 3300, 110, 1, 0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1, 0, 81770000, 185874000, 90563200, 14700400, 943800, 25480, 280, 1
Offset: 0
Examples
[1] [0, 1] [0, 2, 1] [0, 10, 10, 1] [0, 100, 140, 28, 1] [0, 1700, 2900, 840, 60, 1] [0, 44200, 85800, 31460, 3300, 110, 1] [0, 1635400, 3476200, 1501500, 203060, 10010, 182, 1]
Links
- Peter Luschny, The P-transform.
Crossrefs
Programs
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Maple
T := proc(n,k) option remember; if n=k then return 1 fi; if k<0 or k>n then return 0 fi; T(n-1,k-1)+((n-1)^2+k^2)*T(n-1,k) end: seq(seq(T(n,k), k=0..n), n=0..8); # Alternatively with the P-transform (cf. A269941): A268434_row := n -> PTrans(n, n->`if`(n=1,1, ((n-1)^2+1)/(n*(4*n-2))), (n,k)->(-1)^k*(2*n)!/(2*k)!): seq(print(A268434_row(n)), n=0..8);
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Mathematica
T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^2 + k^2)*T[n-1, k]; T[, ] = 0; Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 20 2017 *)
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Sage
#cached_function def T(n, k): if n==k: return 1 if k<0 or k>n: return 0 return T(n-1, k-1)+((n-1)^2+k^2)*T(n-1, k) for n in range(8): print([T(n, k) for k in (0..n)]) # Alternatively with the function PtransMatrix (cf. A269941): PtransMatrix(8, lambda n: 1 if n==1 else ((n-1)^2+1)/(n*(4*n-2)), lambda n, k: (-1)^k*factorial(2*n)/factorial(2*k))
Formula
T(n,k) = (-1)^k*((2*n)!/(2*k)!)*P[n,k](s(n)) where P is the P-transform and s(n) = ((n-1)^2+1)/(n*(4*n-2)). The P-transform is defined in the link. Compare also the Sage and Maple implementations below.
T(n,1) = Product_{k=1..n} (k-1)^2+1 for n>=1 (cf. A101686).
T(n,n-1) = (n-1)*n*(2*n-1)/3 for n>=1 (cf. A006331).
Row sums: A269938.
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