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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.
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%I #11 Nov 29 2023 05:58:26

%S 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,

%T 35,6,1,3445,2863,1449,595,175,45,7,1,14897,12104,6692,2744,950,248,

%U 56,8,1,63313,53769,29772,13236,4686,1422,336,68,9,1

%N 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.

%C The convolution triangle of A091147. - _Peter Luschny_, Oct 07 2022

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

%F 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).

%e [0] 1;

%e [1] 1, 1;

%e [2] 5, 2, 1;

%e [3] 13, 11, 3, 1;

%e [4] 57, 36, 18, 4, 1;

%e [5] 201, 165, 70, 26, 5, 1;

%e [6] 861, 646, 339, 116, 35, 6, 1;

%e [7] 3445, 2863, 1449, 595, 175, 45, 7, 1;

%e [8] 14897, 12104, 6692, 2744, 950, 248, 56, 8, 1;

%e [9] 63313, 53769, 29772, 13236, 4686, 1422, 336, 68, 9, 1.

%p 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):

%p for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

%p # Uses function PMatrix from A357368. Adds a row and column above and to the right.

%p PMatrix(10, n -> simplify(hypergeom([1/2-n/2, 1-n/2], [2], 16))); # _Peter Luschny_, Oct 07 2022

%t (* Uses function PMatrix from A357368 *)

%t nmax = 9;

%t M = PMatrix[nmax+2, HypergeometricPFQ[{1/2 - #/2, 1 - #/2}, {2}, 16]&];

%t T[n_, k_] := M[[n+2, k+2]];

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2023, after _Peter Luschny_ *)

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

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, May 25 2021