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A136489 Triangle T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).

%I A136489 #14 Aug 01 2023 07:09:13
%S A136489 1,1,1,1,4,1,1,5,5,1,1,8,10,8,1,1,9,18,18,9,1,1,12,27,40,27,12,1,1,13,
%T A136489 39,67,67,39,13,1,1,16,52,112,134,112,52,16,1,1,17,68,164,246,246,164,
%U A136489 68,17,1,1,20,85,240,410,504,410,240,85,20,1
%N A136489 Triangle T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).
%H A136489 G. C. Greubel, <a href="/A136489/b136489.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A136489 T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).
%F A136489 Sum_{k=0..n} T(n, k) = A122746(n).
%F A136489 From _G. C. Greubel_, Aug 01 2023: (Start)
%F A136489 T(n, k) = 2*A007318(n, k) - A051159(n, k).
%F A136489 T(n, k) = T(n-1, k) + T(n-1, k-1) if k is even.
%F A136489 T(n, n-k) = T(n, k).
%F A136489 T(n, n-1) = A042948(n).
%F A136489 Sum_{k=0..n} (-1)^k * T(n, k) = 2*[n=0] - A077957(n). (End)
%e A136489 First few rows of the triangle are:
%e A136489   1;
%e A136489   1,   1;
%e A136489   1,   4,   1;
%e A136489   1,   5,   5,   1;
%e A136489   1,   8,  10,   8,   1;
%e A136489   1,   9,  18,  18,   9,   1;
%e A136489   1,  12,  27,  40,  27,  12,   1;
%e A136489   1,  13,  39,  67,  67,  39,  13,   1;
%e A136489   1,  16,  52, 112, 134, 112,  52,  16,   1;
%e A136489   1,  17,  68, 164, 246, 246, 164,  68,  17,   1;
%e A136489   ...
%t A136489 T[n_, k_]:= 2*Binomial[n,k] -Binomial[Mod[n,2], Mod[k,2]]*Binomial[Floor[n/2], Floor[k/2]];
%t A136489 Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Aug 01 2023 *)
%o A136489 (Magma)
%o A136489 A136489:= func< n,k | 2*Binomial(n,k) - Binomial(n mod 2, k mod 2)*Binomial(Floor(n/2), Floor(k/2)) >;
%o A136489 [A136489(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 01 2023
%o A136489 (SageMath)
%o A136489 def A136489(n,k): return 2*binomial(n,k) - binomial(n%2, k%2)*binomial(n//2, k//2)
%o A136489 flatten([[A136489(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Aug 01 2023
%Y A136489 Cf. A034851, A042948, A077957, A122746 (row sums).
%K A136489 nonn,tabl,easy
%O A136489 0,5
%A A136489 _Gary W. Adamson_, Jan 01 2008