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

0,5

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.

Links

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

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.

Example

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;

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 *)

Prog

(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))}

Crossrefs

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

Sequence in context: A121424 A214978 A295380 * A209419 A119011 A340440

Adjacent sequences: A093765 A093766 A093767 * A093769 A093770 A093771

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 16 2004

Status

approved