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A059346 Difference array of Catalan numbers A000108 read by antidiagonals. 9
1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 3, 4, 6, 9, 14, 6, 9, 13, 19, 28, 42, 15, 21, 30, 43, 62, 90, 132, 36, 51, 72, 102, 145, 207, 297, 429, 91, 127, 178, 250, 352, 497, 704, 1001, 1430, 232, 323, 450, 628, 878, 1230, 1727, 2431, 3432, 4862, 603, 835, 1158, 1608, 2236, 3114 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

LINKS

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

F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999), 73-112.

Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

FORMULA

T(n, k) = (-1)^(n-k)*binomial(2*k,k)/(k+1)*hypergeometric([k-n, k+1/2],[k+2], 4). - Peter Luschny, Aug 16 2012

EXAMPLE

Triangle starts:

1;

0, 1;

1, 1, 2;

1, 2, 3, 5;

3, 4, 6, 9, 14;

MAPLE

# Uses floating point, precision might to have be adjusted.

C := n -> binomial(2*n, n)/(n+1);

H := (n, k) -> hypergeom([k-n, k+1/2], [k+2], 4);

T := (n, k) -> (-1)^(n-k)*C(k)*H(n, k);

seq(print(seq(round(evalf(T(n, k), 32)), k=0..n)), n=0..7); # Peter Luschny, Aug 16 2012

MATHEMATICA

max = 11; t = Table[ Differences[ Table[ CatalanNumber[k], {k, 0, max}], n], {n, 0, max}]; Flatten[ Table[t[[n-k+1, k]], {n, 1, max}, {k, 1, n}]] (* Jean-François Alcover, Nov 15 2011 *)

PROG

(Sage)

def T(n, k) :

    if k > n : return 0

    if n == k : return binomial(2*n, n)/(n+1)

    return T(n-1, k) - T(n, k+1)

A059346 = lambda n, k: (-1)^(n-k)*T(n, k)

for n in (0..5): [A059346(n, k) for k in (0..n)] # Peter Luschny, Aug 16 2012

CROSSREFS

Top row is A000108, leading diagonals give A005043, A001006, A005554. Row sums are A106640.

Cf. A000108, A000245, A026012, A033434, A106534.

Sequence in context: A117673 A107946 A054502 * A259439 A274491 A076492

Adjacent sequences:  A059343 A059344 A059345 * A059347 A059348 A059349

KEYWORD

nonn,easy,nice,tabl

AUTHOR

N. J. A. Sloane, Jan 27 2001

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001

STATUS

approved

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Last modified February 23 11:36 EST 2018. Contains 299579 sequences. (Running on oeis4.)