A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths.

1, 1, 1, 1, 2, 3, 1, 3, 8, 6, 1, 4, 15, 20, 20, 1, 5, 24, 45, 75, 50, 1, 6, 35, 84, 189, 210, 175, 1, 7, 48, 140, 392, 588, 784, 490, 1, 8, 63, 216, 720, 1344, 2352, 2352, 1764, 1, 9, 80, 315, 1215, 2700, 5760, 7560, 8820, 5292, 1, 10, 99, 440, 1925, 4950, 12375, 19800

Offset: 0

Author

Paul D. Hanna, Apr 16 2004

Keywords

• nonn
• tabl

Comments

Suggested by Bozydar Dubalski (slawb(AT)atr.bydgoszcz.pl). Related to walks on a square lattice: main diagonal forms A005558, secondary diagonals form A005559, A005560, A005561, A005562, A005563.

Apparently row-reversed version of A052174. - R. J. Mathar, Feb 03 2025

Examples

			1;
1, 1;
1, 2, 3;
1, 3, 8, 6;
1, 4, 15, 20, 20;
1, 5, 24, 45, 75, 50;
1, 6, 35, 84, 189, 210, 175;
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Cf. A088459, A005558-A005562, A005563 (column 3), A005564 (column 4), A005565 (column 5), A005566 (row sums).

Programs

Maple

A093768 := proc(n,k)
    if k = 0 then
        A088459(n,k);
    else
        A088459(n,k)-A088459(n,k-1);
    end if;
end proc:
seq(seq(A093768(n,k),k=0..n-1),n=1..10) ; # R. J. Mathar, Apr 02 2017

Mathematica

T[n_, k_] := Binomial[n + 1, Ceiling[k/2]]*Binomial[n, Floor[k/2]] - Binomial[n + 1, Ceiling[(k - 1)/2]]*Binomial[n, Floor[(k - 1)/2]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 25 2017 *)

PARI

{T(n,k) =binomial(n+1,ceil(k/2))*binomial(n,floor(k/2)) -binomial(n+1,ceil((k-1)/2))*binomial(n,floor((k-1)/2))}

Formula

T(n, k) = C(n+1, ceiling(k/2))*C(n, floor(k/2)) - C(n+1, ceiling((k-1)/2))*C(n, floor((k-1)/2)) for n>=k>=0.