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.

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, 509, 903, 903, 1, 13, 83, 347, 1061, 2473, 4279, 4279, 1, 15, 111, 541, 1949, 5483, 12235, 20793, 20793, 1, 17, 143, 795, 3285, 10717, 28435, 61463, 103049, 103049

Offset: 0

Author

Johannes Fischer (Fischer(AT)informatik.uni-tuebingen.de), Sep 26 2008

Keywords

• nice
• nonn
• tabl

Examples

			First few rows of the triangle:
  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, 509, 903, 903
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• Johannes Fischer and Volker Heun, Finding Range Minima in the Middle: Approximations and Applications, Math. Comput. Sci. 3(1): 17-30 (2010). See Eq. (3.3)
• Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 8.

Crossrefs

Super-Catalan numbers or little Schroeder numbers (cf. A001003) appear on the diagonal.

Generalizes the Catalan triangle (A009766) and hence the ballot Numbers.

Cf. A033877 for a similar triangle derived from the large Schroeder numbers (A006318).

Cf. A010683 (row sums), A186826 (rows reversed).

Cf. A239204, A247623.

Programs

Haskell

a144944 n k = a144944_tabl !! n !! k
a144944_row n = a144944_tabl !! n
a144944_tabl = iterate f [1] where
   f us = vs ++ [last vs] where
     vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us
-- Reinhard Zumkeller, May 11 2013

Mathematica

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 *)

SageMath

@CachedFunction
def t(n,k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (kG. C. Greubel, Mar 11 2023

Formula

From G. C. Greubel, Mar 11 2023: (Start)

Sum_{k=0..n} T(n, k) = A010683(n).

Sum_{k=0..n} (-1)^k*T(n, k) = A239204(n-2).

Sum_{k=0..floor(n/2)} T(n-k, k) = A247623(n). (End)