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A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.

%I A144944 #35 Jun 28 2025 11:15:22
%S A144944 1,1,1,1,3,3,1,5,11,11,1,7,23,45,45,1,9,39,107,197,197,1,11,59,205,
%T A144944 509,903,903,1,13,83,347,1061,2473,4279,4279,1,15,111,541,1949,5483,
%U A144944 12235,20793,20793,1,17,143,795,3285,10717,28435,61463,103049,103049
%N A144944 Super-Catalan triangle (read by rows) = triangular array associated with little Schroeder numbers (read by rows): T(0,0)=1, T(p,q) = T(p,q-1) if 0 < p = q, T(p,q) = T(p,q-1) + T(p-1,q) + T(p-1,q-1) if -1 < p < q and T(p,q) = 0 otherwise.
%H A144944 Reinhard Zumkeller, <a href="/A144944/b144944.txt">Rows n = 0..125 of triangle, flattened</a>
%H A144944 Johannes Fischer and Volker Heun, <a href="https://doi.org/10.1007/s11786-009-0007-8">Finding Range Minima in the Middle: Approximations and Applications</a>, Math. Comput. Sci. 3(1): 17-30 (2010). See Eq. (3.3)
%H A144944 Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 8.
%F A144944 From _G. C. Greubel_, Mar 11 2023: (Start)
%F A144944 Sum_{k=0..n} T(n, k) = A010683(n).
%F A144944 Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2).
%F A144944 Sum_{k=0..floor(n/2)} T(n-k, k) = A247623(n). (End)
%e A144944 First few rows of the triangle:
%e A144944   1
%e A144944   1,  1
%e A144944   1,  3,  3
%e A144944   1,  5, 11,  11
%e A144944   1,  7, 23,  45,  45
%e A144944   1,  9, 39, 107, 197, 197
%e A144944   1, 11, 59, 205, 509, 903, 903
%t A144944 t[_, 0]=1; t[p_, p_]:= t[p, p]= t[p, p-1]; t[p_, q_]:= t[p, q]= t[p, q-1] + t[p-1, q] + t[p-1, q-1]; Flatten[Table[ t[p, q], {p,0,6}, {q,0, p}]] (* _Jean-François Alcover_, Dec 19 2011 *)
%o A144944 (Haskell)
%o A144944 a144944 n k = a144944_tabl !! n !! k
%o A144944 a144944_row n = a144944_tabl !! n
%o A144944 a144944_tabl = iterate f [1] where
%o A144944    f us = vs ++ [last vs] where
%o A144944      vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us
%o A144944 -- _Reinhard Zumkeller_, May 11 2013
%o A144944 (SageMath)
%o A144944 @CachedFunction
%o A144944 def t(n,k):
%o A144944     if (k<0 or k>n): return 0
%o A144944     elif (k==0): return 1
%o A144944     elif (k<n-1): return t(n,k-1) + t(n-1,k) + t(n-1,k-1)
%o A144944     else: return -t(n,n-2)
%o A144944 def T(n,k): return t(n+2,k)
%o A144944 flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 11 2023
%Y A144944 Super-Catalan numbers or little Schroeder numbers (cf. A001003) appear on the diagonal.
%Y A144944 Generalizes the Catalan triangle (A009766) and hence the ballot Numbers.
%Y A144944 Cf. A033877 for a similar triangle derived from the large Schroeder numbers (A006318).
%Y A144944 Cf. A010683 (row sums), A186826 (rows reversed).
%Y A144944 Cf. A239204, A247623.
%K A144944 nice,nonn,tabl
%O A144944 0,5
%A A144944 Johannes Fischer (Fischer(AT)informatik.uni-tuebingen.de), Sep 26 2008