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Triangle T(n,k), read by rows, given by (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
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%I #24 Jan 26 2020 01:05:43

%S 1,0,1,0,1,2,0,2,4,3,0,3,9,10,5,0,5,18,28,22,8,0,8,35,68,74,45,13,0,

%T 13,66,154,210,177,88,21,0,21,122,331,541,574,397,167,34,0,34,222,686,

%U 1302,1656,1446,850,310,55

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

%C Row sums: A133494.

%F Sum_{k=0..n} T(n,k)*x^k = A033999(n), A000007(n), A133494(n) for x = -1, 0, 1 respectively.

%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), for n > 2, T(0,0) = T(1,1) = T(2,1) = 1, T(1,0) = T(2,0) = 0, T(2,2) = 2.

%F T(n+1,n) = A004798(n), T(n,n) = T(n+1,1) = A000045(n+1).

%F T(n,k) = A209138(n,k-1) for k >= 1. - _Philippe Deléham_, Apr 11 2012

%F G.f.: (-1 + x^2*y + x + x^2)/(-1 + x^2*y + x + x^2 + x*y + x^2*y^2). - _R. J. Mathar_, Aug 11 2015

%e Triangle begins:

%e 1;

%e 0, 1;

%e 0, 1, 2;

%e 0, 2, 4, 3;

%e 0, 3, 9, 10, 5;

%e 0, 5, 18, 28, 22, 8;

%e 0, 8, 35, 68, 74, 45, 13;

%e From _Philippe Deléham_, Apr 11 2012: (Start)

%e Triangle in A209138 begins:

%e 1;

%e 1, 2;

%e 2, 4, 3;

%e 3, 9, 10, 5;

%e 5, 18, 28, 22, 8;

%e 8, 35, 68, 74, 45, 13; (End)

%t nmax = 9; T[n_, n_] := Fibonacci[n+1]; T[_, 0] = 0; T[n_, 1] := Fibonacci[n]; T[n_, k_] /; 1 < k < n := T[n, k] = T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k] + T[n - 2, k - 1] + T[n - 2, k - 2]; T[_, _] = 0;

%t Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2017 *)

%Y Cf. A000007, A000045, A000244, A004798, A033999, A084938, A133494, A209138.

%K easy,nonn,tabl

%O 0,6

%A _Philippe Deléham_, Jan 22 2012

%E Corrected by _Jean-François Alcover_, Jun 20 2017