A144944 - OEIS

%I #28 Mar 11 2023 08:16:39

%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 [math.RA], 2015. See Fig. 8.

%F From _G. C. Greubel_, Mar 11 2023: (Start)

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

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

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

%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-François 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

%o (SageMath)

%o @CachedFunction

%o def t(n,k):

%o     if (k<0 or k>n): return 0

%o     elif (k==0): return 1

%o     elif (k<n-1): return t(n,k-1) + t(n-1,k) + t(n-1,k-1)

%o     else: return -t(n,n-2)

%o def T(n,k): return t(n+2,k)

%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Mar 11 2023

%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).

%Y Cf. A239204, A247623.

%K nice,nonn,tabl

%O 0,5

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