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A128899 Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).

%I A128899 #43 Mar 24 2025 10:21:54
%S A128899 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 A128899 0,429,572,429,208,65,12,1,0,1430,2002,1638,910,350,90,14,1,0,4862,
%U A128899 7072,6188,3808,1700,544,119,16,1,0,16796,25194,23256,15504,7752,2907,798,152,18,1
%N A128899 Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).
%C A128899 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 A128899 T is the convolution triangle of the Catalan numbers (see A357368). - _Peter Luschny_, Oct 19 2022
%H A128899 Michael De Vlieger, <a href="/A128899/b128899.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150, flattened)
%H A128899 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 A128899 Paul Barry, <a href="https://arxiv.org/abs/1912.11845">Chebyshev moments and Riordan involutions</a>, arXiv:1912.11845 [math.CO], 2019.
%F A128899 T(n,k) = A039598(n-1,k-1) for n >= 1, k >= 1; T(n,0)=0^n.
%F A128899 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 A128899 T(n,k) + T(n,k+1) = A039599(n,k). - _Philippe Deléham_, Sep 12 2007
%e A128899 Triangle begins:
%e A128899   1;
%e A128899   0,     1;
%e A128899   0,     2,     1;
%e A128899   0,     5,     4,     1;
%e A128899   0,    14,    14,     6,     1;
%e A128899   0,    42,    48,    27,     8,    1;
%e A128899   0,   132,   165,   110,    44,   10,    1;
%e A128899   0,   429,   572,   429,   208,   65,   12,  1;
%e A128899   0,  1430,  2002,  1638,   910,  350,   90,  14,   1;
%e A128899   0,  4862,  7072,  6188,  3808, 1700,  544, 119,  16,  1;
%e A128899   0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1;
%e A128899   ...
%p A128899 # Uses function PMatrix from A357368.
%p A128899 PMatrix(10, n -> binomial(2*n,n)/(n+1)); # _Peter Luschny_, Oct 19 2022
%t A128899 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 A128899 Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* _Jean-François Alcover_, Jun 14 2019 *)
%o A128899 (Sage)
%o A128899 @cached_function
%o A128899 def T(k,n):
%o A128899     if k==n: return 1
%o A128899     if k==0: return 0
%o A128899     return sum(catalan_number(i)*T(k-1,n-i) for i in (1..n-k+1))
%o A128899 A128899 = lambda n,k: T(k,n)
%o A128899 for n in (0..10): print([A128899(n,k) for k in (0..n)]) # _Peter Luschny_, Mar 12 2016
%o A128899 (PARI) T(n, k) = binomial(2*n-2, n-k)-binomial(2*n-2, n-k-2); \\ _Seiichi Manyama_, Mar 24 2025
%Y A128899 Row sums give A088218.
%Y A128899 Cf. A000108, A008313, A039598, A039599, A050166.
%K A128899 nonn,tabl
%O A128899 0,5
%A A128899 _Philippe Deléham_, Apr 21 2007
%E A128899 Typo in data corrected by _Jean-François Alcover_, Jun 14 2019