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A130020 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 .

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