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

Author

Peter Luschny, Mar 07 2016

Keywords

• nonn
• tabl

Comments

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

Cf. A038207 (order 0), A111596 (order 1), A269946 (order 3).

Cf. A036969, A105278, A204579, A269938, A269941, A269944, A269945.

Programs

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);

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 *)

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,k) = Sum_{j=k..n} A269944(n,j)*A269945(j,k).

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.