A158196
Expansion of (1-x^2*c(x)^4)/(1-3*x*c(x)^2), c(x) the g.f. of A000108.
Original entry on oeis.org
1, 3, 14, 71, 370, 1950, 10332, 54895, 292106, 1555706, 8289732, 44186710, 235575028, 1256093084, 6698073528, 35719158591, 190488112122, 1015885525794, 5417869631028, 28894620083346, 154102115782812
Offset: 0
A158197
Expansion of (1-x^2*c(x)^4)/(1-4*x*c(x)^2), c(x) the g.f. of A000108.
Original entry on oeis.org
1, 4, 23, 140, 866, 5388, 33603, 209796, 1310510, 8188328, 51169094, 319779544, 1998527188, 12490460620, 78064190235, 487896926580, 3049340393430, 19058321475960, 119114304522450, 744463650984360, 4652895041524380
Offset: 0
A293944
Triangle read by rows related to Catalan triangle A009766.
Original entry on oeis.org
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
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,
...
- 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).
-
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
-
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 *)
Showing 1-3 of 3 results.
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