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A216154
Triangle read by rows, T(n,k) n>=0, k>=0, generalization of A000255.
1
1, 1, 1, 3, 4, 1, 11, 21, 9, 1, 53, 128, 78, 16, 1, 309, 905, 710, 210, 25, 1, 2119, 7284, 6975, 2680, 465, 36, 1, 16687, 65821, 74319, 35035, 7945, 903, 49, 1, 148329, 660064, 857836, 478464, 133630, 19936, 1596, 64, 1, 1468457, 7275537, 10690812, 6879684, 2279214, 419958, 44268, 2628, 81, 1
OFFSET
0,4
FORMULA
Recurrence: T(0,0)=1, T(0,k)=0 for k>0 and for n>=1 T(n,k) = T(n-1,k-1)+(1+2*k)*T(n-1,k)+(k+1)*(k+2)*T(n-1,k+1).
Let Z(n, k) = Sum_{j=0..n} C(-j, -n)*L(j, k) where L denotes the unsigned Lah numbers A271703. Then T(n, k) = Z(n+1, k+1). - Peter Luschny, Apr 13 2016
EXAMPLE
1,
1, 1,
3, 4, 1,
11, 21, 9, 1,
53, 128, 78, 16, 1,
309, 905, 710, 210, 25, 1,
2119, 7284, 6975, 2680, 465, 36, 1,
16687, 65821, 74319, 35035, 7945, 903, 49, 1,
148329, 660064, 857836, 478464, 133630, 19936, 1596, 64, 1,
MAPLE
A216154 := proc(n, k) local L, Z;
L := (n, k) -> `if`(k<0 or k>n, 0, (n-k)!*C(n, n-k)*C(n-1, n-k)):
Z := (n, k) -> add(C(-j, -n)*L(j, k), j=0..n);
Z(n+1, k+1) end:
seq(seq(A216154(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 13 2016
MATHEMATICA
T[0, 0] = 1; T[0, _] = 0; T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k-1] + (2k+1) T[n-1, k] + (k+1) (k+2) T[n-1, k+1]; T[_, _] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
PROG
(Sage)
def A216154_triangle(dim):
M = matrix(ZZ, dim, dim)
for n in (0..dim-1): M[n, n] = 1
for n in (1..dim-1):
for k in (0..n-1):
M[n, k] = M[n-1, k-1]+(1+2*k)*M[n-1, k]+(k+1)*(k+2)*M[n-1, k+1]
return M
A216154_triangle(9)
CROSSREFS
A000255 (col. 0), A110450 (diag. n,n-2).
Sequence in context: A172094 A114608 A154602 * A325174 A109956 A123319
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Sep 19 2012
STATUS
approved