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.
Zhousheng Mei, Suijie Wang, Pattern Avoidance of Generalized Permutations, arXiv:1804.06265 [math.CO], 2018.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024. See p. 36.
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
Array starts:
1 1 2 5 14 42 132 429
0 1 3 9 28 90 297 1001
1 2 6 19 62 207 704 2431
1 4 13 43 145 497 1727 6071
3 9 30 102 352 1230 4344 15483
6 21 72 250 878 3114 11139 40143
15 51 178 628 2236 8025 29004 105477
36 127 450 1608 5789 20979 76473 280221
91 323 1158 4181 15190 55494 203748 751422
232 835 3023 11009 40304 148254 547674 2031054
603 2188 7986 29295 107950 399420 1483380 5527750
Triangle starts:
1;
0, 1;
1, 1, 2;
1, 2, 3, 5;
3, 4, 6, 9, 14;
MAPLE
T := (n, k) -> (-1)^(n-k)*binomial(2*k, k)*hypergeom([k-n, k+1/2], [k+2], 4)/(k+1): seq(seq(simplify(T(n, k)), k=0..n), n=0..10);
# Peter Luschny, Aug 16 2012, updated May 25 2021
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
KEYWORD
AUTHOR
N. J. A. Sloane, Jan 27 2001
EXTENSIONS
More terms from Larry Reeves (larryr(AT)acm.org), Feb 16 2001
STATUS
approved