OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Sum_{k=0..n} T(n, k) = A122746(n).
From G. C. Greubel, Aug 01 2023: (Start)
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)
EXAMPLE
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;
...
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Gary W. Adamson, Jan 01 2008
STATUS
approved