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Triangle T(n,k), read by rows, given by (2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
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%I #10 Aug 08 2015 10:30:40

%S 1,2,2,5,8,3,12,27,20,5,29,84,91,44,8,70,248,352,251,90,13,169,708,

%T 1240,1164,618,176,21,408,1973,4106,4771,3344,1414,334,34,985,5400,

%U 13010,18000,15645,8748,3073,620,55

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

%C Row sums are powers of 4 (A000302).

%F G.f.: (1+y*x)/(1-(y+2)*x-(y+1)^2*x^2).

%F T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + 2*T(n-2,k-1) + T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k<0 or if k>n.

%F Sum_{k, 0<=k<=n} T(n,k)*x^k = (-1)^n*A159612(n+1), (-1)^n*A000034(n), A000007(n), A000129(n+1), A000302(n) for x = -3, -2, -1, 0, 1 respectively.

%F T(n,0) = A000129(n+1), T(n,n) = A000045(n+2), T(n+1,n) = 2*A004798(n+1).

%e Triangle begins :

%e 1

%e 2, 2

%e 5, 8, 3

%e 12, 27, 20, 5

%e 29, 84, 91, 44, 8

%e 70, 248, 352, 251, 90, 13

%e 169, 708, 1240, 1164, 618, 176, 21

%e 408, 1973, 4106, 4771, 3344, 1414, 334, 34

%e 985, 5400, 13010, 18000, 15645, 8748, 3073, 620, 55

%e 2378, 14574, 39880, 63966, 66282, 46014, 21400, 6429, 1132, 89

%e 5741, 38896, 119129, 217232, 261185, 216348, 125028, 49772, 13061, 2040, 144

%Y Cf. A000045, A000129, A000302, A261056 (2nd column).

%K easy,nonn,tabl

%O 0,2

%A _Philippe Deléham_, Mar 26 2012