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

0,8

Comments

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

Links

Table of n, a(n) for n=0..65.

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

Example

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.

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<k<n := T[n, k] =T[n-1, k-1] + T[n-1, k] + T[n-1, k+1]; T[_, _] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

Prog

(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

Crossrefs

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

Product of A007318 and this sequence is A122897.

Cf. A001006, A007318, A064189.

Sequence in context: A332011 A229762 A062110 * A191348 A198792 A196182

Adjacent sequences: A122893 A122894 A122895 * A122897 A122898 A122899

Keyword

easy,nonn,tabl

Author

Paul Barry, Sep 18 2006

Status

approved