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A093768 Positive first differences of the rows of triangle A088459, which enumerates symmetric Dyck paths. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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 A300866

Adjacent sequences:  A093765 A093766 A093767 * A093769 A093770 A093771

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Apr 16 2004

STATUS

approved

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Last modified December 13 03:41 EST 2019. Contains 329968 sequences. (Running on oeis4.)