%I #26 Oct 01 2022 23:23:08
%S 1,1,0,1,1,0,1,2,2,0,1,3,5,5,0,1,4,9,14,14,0,1,5,14,28,42,42,0,1,6,20,
%T 48,90,132,132,0,1,7,27,75,165,297,429,429,0,1,8,35,110,275,572,1001,
%U 1430,1430,0,1,9,44,154,429,1001,2002,3432,4862,4862,0
%N Triangle T(n,k), 0<=k<=n, read by rows given by [1,0,0,0,0,0,0,...] DELTA [0,1,1,1,1,1,1,...] where DELTA is the operator defined in A084938 .
%C Reflected version of A106566.
%H G. C. Greubel, <a href="/A130020/b130020.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Francesca Aicardi, <a href="https://arxiv.org/abs/2011.14628">Catalan triangle and tied arc diagrams</a>, arXiv:2011.14628 [math.CO], 2020.
%F T(n, k) = A106566(n, n-k).
%F Sum_{k=0..n} T(n,k) = A000108(n).
%F T(n, k) = (n-k)*binomial(n+k-1, k)/n with T(0, 0) = 1. - _Jean-François Alcover_, Jun 14 2019
%F Sum_{k=0..floor(n/2)} T(n-k, k) = A210736(n). - _G. C. Greubel_, Jun 14 2022
%F G.f.: Sum_{n>=0, k>=0} T(n, k)*x^k*z^n = 1/(1 - z*c(x*z)) where c(z) = g.f. of A000108.
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 1, 2, 2, 0;
%e 1, 3, 5, 5, 0;
%e 1, 4, 9, 14, 14, 0;
%e 1, 5, 14, 28, 42, 42, 0;
%e 1, 6, 20, 48, 90, 132, 132, 0;
%e 1, 7, 27, 75, 165, 297, 429, 429, 0;
%e 1, 8, 35, 110, 275, 572, 1001, 1430, 1430, 0;
%e 1, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 0;
%e ...
%t T[n_, k_]:= (n-k)Binomial[n+k-1, k]/n; T[0, 0] = 1;
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* _Jean-François Alcover_, Jun 14 2019 *)
%o (Sage)
%o @CachedFunction
%o def A130020(n, k):
%o if n==k: return add((-1)^j*binomial(n, j) for j in (0..n))
%o return add(A130020(n-1, j) for j in (0..k))
%o for n in (0..10) :
%o [A130020(n, k) for k in (0..n)] # _Peter Luschny_, Nov 14 2012
%o (Magma)
%o A130020:= func< n,k | n eq 0 select 1 else (n-k)*Binomial(n+k-1, k)/n >;
%o [A130020(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 14 2022
%o (PARI) {T(n, k) = if( k<0 || k>=n, n==0 && k==0, binomial(n+k, n) * (n-k)/(n+k))}; /* _Michael Somos_, Oct 01 2022 */
%Y The following are all versions of (essentially) the same Catalan triangle: A009766, A030237, A033184, A047072, A059365, A099039, A106566, this sequence.
%Y Diagonals give A000108, A000245, A002057, A000344, A003517, A000588, A003518, A003519, A001392, ...
%Y Cf. A000108 (Catalan numbers), A106566 (row reversal), A210736.
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Jun 16 2007