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%I #35 Oct 19 2022 11:04:13
%S 1,0,1,0,2,1,0,5,4,1,0,14,14,6,1,0,42,48,27,8,1,0,132,165,110,44,10,1,
%T 0,429,572,429,208,65,12,1,0,1430,2002,1638,910,350,90,14,1,0,4862,
%U 7072,6188,3808,1700,544,119,16,1,0,16796,25194,23256,15504,7752,2907,798,152,18,1
%N Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).
%C Let the sequence A(n) = [0/1, 2/1, 1/2, 3/2, 2/3, 4/3, ...] defined by a(2n)=n/(n+1) and a(2n+1)=(n+2)/(n+1). T(n,k) is the triangle read by rows given by A(n) DELTA A000007 where DELTA is the operator defined in A084938.
%C T is the convolution triangle of the Catalan numbers (see A357368). - _Peter Luschny_, Oct 19 2022
%H Michael De Vlieger, <a href="/A128899/b128899.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H Paul Barry, <a href="https://arxiv.org/abs/1912.01124">A Note on Riordan Arrays with Catalan Halves</a>, arXiv:1912.01124 [math.CO], 2019.
%H Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%F T(n,k) = A039598(n-1,k-1) for n >= 1, k >= 1; T(n,0)=0^n.
%F T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k >= 1, T(n,0)=0^n, T(n,k)=0 if k > n.
%F T(n,k) + T(n,k+1) = A039599(n,k). - _Philippe Deléham_, Sep 12 2007
%e Triangle begins:
%e 1;
%e 0, 1;
%e 0, 2, 1;
%e 0, 5, 4, 1;
%e 0, 14, 14, 6, 1;
%e 0, 42, 48, 27, 8, 1;
%e 0, 132, 165, 110, 44, 10, 1;
%e 0, 429, 572, 429, 208, 65, 12, 1;
%e 0, 1430, 2002, 1638, 910, 350, 90, 14, 1;
%e 0, 4862, 7072, 6188, 3808, 1700, 544, 119, 16, 1;
%e 0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1;
%e ...
%p # Uses function PMatrix from A357368.
%p PMatrix(10, n -> binomial(2*n,n)/(n+1)); # _Peter Luschny_, Oct 19 2022
%t T[n_, n_] := 1; T[_, 0] = 0; T[n_, k_] /; 0 < k < n := T[n, k] = T[n - 1, k - 1] + 2 T[n - 1, k] + T[n - 1, k + 1]; T[_, _] = 0;
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* _Jean-François Alcover_, Jun 14 2019 *)
%o (Sage)
%o @cached_function
%o def T(k,n):
%o if k==n: return 1
%o if k==0: return 0
%o return sum(catalan_number(i)*T(k-1,n-i) for i in (1..n-k+1))
%o A128899 = lambda n,k: T(k,n)
%o for n in (0..10): print([A128899(n,k) for k in (0..n)]) # _Peter Luschny_, Mar 12 2016
%Y Cf. A000108, A039598.
%K nonn,tabl
%O 0,5
%A _Philippe Deléham_, Apr 21 2007
%E Typo in data corrected by _Jean-François Alcover_, Jun 14 2019
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