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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. 4

%I

%S 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 509,903,903,1,13,83,347,1061,2473,4279,4279,1,15,111,541,1949,5483,

%U 12235,20793,20793,1,17,143,795,3285,10717,28435,61463,103049,103049

%N 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 Reinhard Zumkeller, <a href="/A144944/b144944.txt">Rows n = 0..125 of triangle, flattened</a>

%H Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757, 2015. See Fig. 8

%e First few rows of the triangle:

%e 1

%e 1, 1

%e 1, 3, 3

%e 1, 5, 11, 11

%e 1, 7, 23, 45, 45

%e 1, 9, 39, 107, 197, 197

%e 1, 11, 59, 205, 509, 903, 903

%t 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-Fran├žois Alcover_, Dec 19 2011 *)

%o (Haskell)

%o a144944 n k = a144944_tabl !! n !! k

%o a144944_row n = a144944_tabl !! n

%o a144944_tabl = iterate f [1] where

%o f us = vs ++ [last vs] where

%o vs = scanl1 (+) $ zipWith (+) us $ [0] ++ us

%o -- _Reinhard Zumkeller_, May 11 2013

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

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

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

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

%K nice,nonn,tabl

%O 0,5

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

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Last modified January 21 11:00 EST 2019. Contains 319351 sequences. (Running on oeis4.)