A344557 Triangle read by rows, T(n, k) = 2^(n - k)*M(n, k, 1/2, 1/2), where M(n, k, x, y) is a generalized Motzkin recurrence. T(n, k) for 0 <= k <= n.

1, 1, 1, 5, 2, 1, 13, 11, 3, 1, 57, 36, 18, 4, 1, 201, 165, 70, 26, 5, 1, 861, 646, 339, 116, 35, 6, 1, 3445, 2863, 1449, 595, 175, 45, 7, 1, 14897, 12104, 6692, 2744, 950, 248, 56, 8, 1, 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1

Offset: 0

Author

Peter Luschny, May 25 2021

Keywords

• nonn
• tabl

Comments

The convolution triangle of A091147. - Peter Luschny, Oct 07 2022

Examples

			[0]     1;
[1]     1,     1;
[2]     5,     2,     1;
[3]    13,    11,     3,     1;
[4]    57,    36,    18,     4,    1;
[5]   201,   165,    70,    26,    5,    1;
[6]   861,   646,   339,   116,   35,    6,   1;
[7]  3445,  2863,  1449,   595,  175,   45,   7,  1;
[8] 14897, 12104,  6692,  2744,  950,  248,  56,  8, 1;
[9] 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1.
		

Crossrefs

A091147 (first column), A344558 (row sums).

Programs

Maple

t := proc(n, k) option remember; if n = k then 1 elif k < 0 or n < 0 or k > n then 0 elif k = 0 then t(n-1, 0)/2 + t(n-1, 1) else t(n-1, k-1) + (1/2)*t(n-1, k) + t(n-1, k+1) fi end: T := (n, k) -> 2^(n-k) * t(n, k):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;
# Uses function PMatrix from A357368. Adds a row and column above and to the right.
PMatrix(10, n -> simplify(hypergeom([1/2-n/2, 1-n/2], [2], 16))); # Peter Luschny, Oct 07 2022

Mathematica

(* Uses function PMatrix from A357368 *)
nmax = 9;
M = PMatrix[nmax+2, HypergeometricPFQ[{1/2 - #/2, 1 - #/2}, {2}, 16]&];
T[n_, k_] := M[[n+2, k+2]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023, after Peter Luschny *)

Formula

The generalized Motzkin recurrence M(n, k, x, y) is defined as follows:

If k < 0 or n < 0 or k > n then 0 else if n = 0 then 1 else if k = 0 then x*M(n-1, 0, x, y) + M(n-1, 1, x, y). In all other cases M(n, k, x, y) = M(n-1, k-1, x, y) + y*M(n-1, k, x, y) + M(n-1, k+1, x, y).