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

0,5

Links

Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015. See Fig. 8.

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)

Example

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

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 *)

Prog

(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
(SageMath)
@CachedFunction
def t(n, k):
    if (k<0 or k>n): return 0
    elif (k==0): return 1
    elif (k<n-1): return t(n, k-1) + t(n-1, k) + t(n-1, k-1)
    else: return -t(n, n-2)
def T(n, k): return t(n+2, k)
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Mar 11 2023

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.

Sequence in context: A193823 A071945 A209583 * A137426 A269949 A074456

Adjacent sequences: A144941 A144942 A144943 * A144945 A144946 A144947

Keyword

nice,nonn,tabl

Author

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

Status

approved