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Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice. arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.

Fang, E., Jenkins, J., Lee, Z., Li, D., Lu, E., Miller, S. J., ... & Siktar, J. (2019). <a href="https://arxiv.org/abs/1906.10645">Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice</a>, arXiv preprint arXiv:1906.10645. Also Fib. Q., 58:1 (2020), 208-225.

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T(n, n-2) = 2*A049611(n-1), n >= 2.

T(n, n-3) = 4*A049612(n-2), n >= 3.

T(n, n-4) = 8*A055589(n-3), n >= 4.

T(n, n-6) = 32*A055853(n-5), n >= 6.

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T(n, n-1) = A001792(n-1).

T(2*n, n) = A052141(n).

Sum_{k=0..n} T(n, k) = A003480(n).

From G. C. Greubel, Sep 02 2022: (Start)

T(n, n-k) = T(n, k).

T(n, 0) = T(n, n) = A011782(n).

T(n, n-2) = 2*A049611(n-1), n >= 2.

T(n, n-3) = 4*A049612(n-2), n >= 3.

T(n, n-4) = 8*A055589(n-3), n >= 4.

T(n, n-5) = 16*A055852(n-4), n >= 5.

T(n, n-6) = 32*A055853(n-5), n >= 6

Sum_{k=0..floor(n/2)} T(n, k) = A181306(n). (End)

tT[0, 0] = 1; tT[n_, k_] := 2^(n-k-1)*n!*Hypergeometric2F1[ -k, -k, -n, -1 ] / (k!*(n-k)!); Flatten[ Table[ tT[n, k], {n, 0, 9}, {k, 0, n}]] (* Jean-François Alcover, Feb 01 2012, after Robert Israel *)

(Magma)

A011782:= func< n | n eq 0 select 1 else 2^(n-1) >;

function T(n, k) // T = A059576

if k eq 0 or k eq n then return A011782(n);

else return 2*T(n-1, k-1) + 2*T(n-1, k) - (2 - 0^(n-2))*T(n-2, k-1);

end if; return T;

end function;

[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Sep 02 2022

(SageMath)

def T(n, k): # T = A059576

if (k==0 or k==n): return 1 if (n==0) else 2^(n-1) # A011782

else: return 2*T(n-1, k-1) + 2*T(n-1, k) - (2 - 0^(n-2))*T(n-2, k-1)

flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Sep 02 2022

First diagonals give A000079, A001792.

T(2*n, n) gives A052141. Row sums give A003480.

Cf. A000079, A001792, A003480 (row sums), A052141 (main diagonal).

Cf. A011782, A049611, A049612, A055589, A055852, A055853, A181306.

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