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

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, 39, 67, 67, 39, 13, 1, 1, 16, 52, 112, 134, 112, 52, 16, 1, 1, 17, 68, 164, 246, 246, 164, 68, 17, 1, 1, 20, 85, 240, 410, 504, 410, 240, 85, 20, 1

Offset: 0

Author

Gary W. Adamson, Jan 01 2008

Keywords

• nonn
• tabl
• easy

Examples

			First few rows of the triangle are:
  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,  39,  67,  67,  39,  13,   1;
  1,  16,  52, 112, 134, 112,  52,  16,   1;
  1,  17,  68, 164, 246, 246, 164,  68,  17,   1;
  ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A034851, A042948, A077957, A122746 (row sums).

Programs

Magma

A136489:= func< n,k | 2*Binomial(n,k) - Binomial(n mod 2, k mod 2)*Binomial(Floor(n/2), Floor(k/2)) >;
[A136489(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2023

Mathematica

T[n_, k_]:= 2*Binomial[n,k] -Binomial[Mod[n,2], Mod[k,2]]*Binomial[Floor[n/2], Floor[k/2]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2023 *)

SageMath

def A136489(n,k): return 2*binomial(n,k) - binomial(n%2, k%2)*binomial(n//2, k//2)
flatten([[A136489(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 01 2023

Formula

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

Sum_{k=0..n} T(n, k) = A122746(n).

From G. C. Greubel, Aug 01 2023: (Start)

T(n, k) = 2*A007318(n, k) - A051159(n, k).

T(n, k) = T(n-1, k) + T(n-1, k-1) if k is even.

T(n, n-k) = T(n, k).

T(n, n-1) = A042948(n).

Sum_{k=0..n} (-1)^k * T(n, k) = 2*[n=0] - A077957(n). (End)