A122896 Riordan array (1, (1 - x - sqrt(1 - 2*x - 3*x^2)) / (2*x)), a Riordan array for directed animals. Triangle read by rows.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 5, 3, 1, 0, 9, 12, 9, 4, 1, 0, 21, 30, 25, 14, 5, 1, 0, 51, 76, 69, 44, 20, 6, 1, 0, 127, 196, 189, 133, 70, 27, 7, 1, 0, 323, 512, 518, 392, 230, 104, 35, 8, 1, 0, 835, 1353, 1422, 1140, 726, 369, 147, 44, 9, 1

Offset: 0

Author

Paul Barry, Sep 18 2006

Keywords

• easy
• nonn
• tabl

Comments

Also the convolution triangle of the Motzkin numbers A001006. - Peter Luschny, Oct 08 2022

Examples

			Triangle begins:
[0] 1;
[1] 0,   1;
[2] 0,   1,   1;
[3] 0,   2,   2,   1;
[4] 0,   4,   5,   3,   1;
[5] 0,   9,  12,   9,   4,   1;
[6] 0,  21,  30,  25,  14,   5,   1;
[7] 0,  51,  76,  69,  44,  20,   6,  1;
[8] 0, 127, 196, 189, 133,  70,  27,  7, 1;
[9] 0, 323, 512, 518, 392, 230, 104, 35, 8, 1.
		

Crossrefs

Row sums are A005773, number of directed animals of size n.

Product of A007318 and this sequence is A122897.

Cf. A001006, A007318, A064189.

Programs

Maple

T := proc(n,k) option remember;
if k=0 then return 0^n fi; if k>n then return 0 fi;
T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) end:
for n from 0 to 9 do seq(T(n,k), k=0..n) od; # Peter Luschny, Aug 17 2016
# Uses function PMatrix from A357368.
PMatrix(10, n -> simplify(hypergeom([1 -n/2, -n/2+1/2], [2], 4))); # Peter Luschny, Oct 08 2022

Mathematica

T[n_, n_] = 1; T[, 0] = 0; T[n, k_] /; 0, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

Sage

# uses[riordan_array from A256893]
riordan_array(1, (1-x-sqrt(1-2*x-3*x^2))/(2*x), 11) # Peter Luschny, Aug 17 2016

Formula

Inverse of Riordan array (1, x / (1 + x + x^2)).

T(n+1, k+1) = A064189(n, k). - Philippe Deléham, Apr 21 2007

Riordan array (1, x*m(x)) where m(x) is the g.f. of Motzkin numbers (A001006). - Philippe Deléham, Nov 04 2009