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A204533 Triangle T(n,k), read by rows, given by (0, 1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

%I A204533 #18 Oct 02 2018 05:40:13
%S A204533 1,0,1,0,1,1,0,3,2,1,0,8,7,3,1,0,21,22,12,4,1,0,55,67,43,18,5,1,0,144,
%T A204533 200,147,72,25,6,1,0,377,588,486,271,110,33,7,1,0,987,1708,1566,976,
%U A204533 450,158,42,8,1
%N A204533 Triangle T(n,k), read by rows, given by (0, 1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
%C A204533 Riordan array (1, x*(1-x)^2/(1-3*x+x^2)).
%C A204533 Antidiagonal sums: see A052946.
%H A204533 Michael De Vlieger, <a href="/A204533/b204533.txt">Table of n, a(n) for n = 0..11475</a> (rows 0 <= n <= 150).
%F A204533 Sum_{k=0..n} T(n,k) = A204200(n+1).
%F A204533 T(n,k) = 3*T(n-1,k) + T(n-1,k-1) + T(n-3,k-1) - T(n-2,k) - 2*T(n-2,k-1).
%F A204533 G.f.: (-1 + 3*x - x^2)/(-1 + 3*x - x^2 + x*y - 2*x^2*y + x^3*y). - _R. J. Mathar_, Aug 11 2015
%F A204533 T(n,m) = Sum_{k=0..n-1} C(k,m-1)*C(n-2*m+k,n-k-1), T(0,0)=1. - _Vladimir Kruchinin_, Sep 27 2018
%e A204533 Triangle begins:
%e A204533   1;
%e A204533   0,   1;
%e A204533   0,   1,   1;
%e A204533   0,   3,   2,   1;
%e A204533   0,   8,   7,   3,  1;
%e A204533   0,  21,  22,  12,  4,  1;
%e A204533   0,  55,  67,  43, 18,  5, 1;
%e A204533   0, 144, 200, 147, 72, 25, 6, 1;
%t A204533 Table[Sum[Binomial[k, m - 1] Binomial[n - 2 m + k, n - k - 1], {k, 0, n - 1}] + Boole[n == m == 0], {n, 0, 9}, {m, 0, n}] // Flatten (* _Michael De Vlieger_, Sep 26 2018 *)
%o A204533 (Maxima)
%o A204533 T(n,m):= if n=0 and m=0 then 1 else sum(binomial(k,m-1)*binomial(n-2*m+k,n-k-1),k,0,n-1); /* _Vladimir Kruchinin_, Sep 27 2018 */
%o A204533 (PARI) T(n,k) = if ((n==0) && (k==0), 1, sum(i=0, n-1, binomial(i,k-1)*binomial(n-2*k+i,n-i-1))); \\ _Michel Marcus_, Sep 27 2018
%Y A204533 Cf. diagonals: A000007, A088305, A000012, A001477, A055998.
%K A204533 nonn,tabl
%O A204533 0,8
%A A204533 _Philippe Deléham_, Jan 16 2012