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A293944
Triangle read by rows related to Catalan triangle A009766.
1
1, 1, 1, 2, 3, 2, 5, 9, 9, 5, 14, 28, 34, 28, 14, 42, 90, 123, 123, 90, 42, 132, 297, 440, 497, 440, 297, 132, 429, 1001, 1573, 1935, 1935, 1573, 1001, 429, 1430, 3432, 5642, 7397, 8068, 7397, 5642, 3432, 1430, 4862, 11934, 20332, 28014, 32636, 32636, 28014, 20332, 11934
OFFSET
0,4
LINKS
Laurent Méhats, Lutz Straßburger, Non-crossing Tree Realizations of Ordered Degree Sequences, Pages 211-227 in Logical Aspects of Computational Linguistics. Celebrating 20 Years of LACL (1996-2016), 9th International Conference, LACL 2016, Nancy, France, December 5-7, 2016, Proceedings, Lecture Notes in Computer Science book series (LNCS, volume 10054). See Eq. (7).
EXAMPLE
Triangle begins:
1,
1,1,
2,3,2,
5,9,9,5,
14,28,34,28,14,
42,90,123,123,90,42,
132,297,440,497,440,297,132,
...
MAPLE
A000108 := proc(q)
if q <0 then
0;
else
binomial(2*q, q)/(1+q) ;
end if;
end proc:
R := proc(q, s)
option remember;
local a, j, l ;
if q= 0 then
A000108(s) ;
elif s = 0 then
A000108(q) ;
else
a := 0 ;
for j from 0 to q do
for l from 0 to s do
if j+l-1 >= 0 then
a := a+A000108(j+l-1) *procname(q-j, s-l) ;
end if;
end do:
end do:
end if;
end proc:
A293944 := proc(n, k)
R(n-k, k) ;
end proc:
seq(seq(A293944(n, k), k=0..n), n=0..12) ; # R. J. Mathar, Nov 02 2017
MATHEMATICA
R[q_, s_] := R[q, s] = Module[{a, j, l}, If[q == 0, CatalanNumber[s], If[s == 0, CatalanNumber[q], a = 0; For[j = 0, j <= q, j++, For[l = 0, l <= s , l++, If[j + l - 1 >= 0, a = a + CatalanNumber[j + l - 1] R[q - j, s - l]] ]]]] /. Null -> a];
T [n_, k_] := R[n - k, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 07 2020, after R. J. Mathar *)
CROSSREFS
Cf. A009766, A000108 (1st column), A000245 (2nd column), A120989 (3rd), A090317 (row sums).
Sequence in context: A367859 A349789 A337354 * A050159 A147294 A296662
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Oct 21 2017
EXTENSIONS
More terms from R. J. Mathar, Nov 02 2017
STATUS
approved