OFFSET
0,5
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
From G. C. Greubel, Dec 25 2025: (Start)
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n,k) = A007283(n-1) - (1/2)*[n=0] - [n=1] - 2*[n=2].
Sum_{k=0..floor(n/2)} T(n-k, k) = A244472(n) + [n=0].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = 3*A010892(n) - 2*([n=0] + [n=1] - [n=3]). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 5, 5, 1;
1, 6, 10, 6, 1;
1, 7, 16, 16, 7, 1;
1, 8, 23, 32, 23, 8, 1;
1, 9, 31, 55, 55, 31, 9, 1;
1, 10, 40, 86, 110, 86, 40, 10, 1;
1, 11, 50, 126, 196, 196, 126, 50, 11, 1;
1, 12, 61, 176, 322, 392, 322, 176, 61, 12, 1;
MATHEMATICA
Table[Binomial[n, m] + If[n>2, 2*Binomial[n-2, m-1], 0], {n, 0, 12}, {m, 0, n}]//Flatten
PROG
(Magma)
A147644:= function(n, k)
if n le 2 then return Binomial(n, k);
else return Binomial(n, k) + 2*Binomial(n-2, k-1);
end if; end function;
[A147644(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 25 2025
(SageMath)
def A147644(n, k):
if n<3: return binomial(n, k)
else: return binomial(n, k) + 2*binomial(n-2, k-1)
print(flatten([[A147644(n, k) for k in range(n+1)] for n in range(13)])) # G. C. Greubel, Dec 25 2025
CROSSREFS
Sums: A000007 (antidiagonal).
KEYWORD
AUTHOR
Roger L. Bagula, Nov 09 2008
EXTENSIONS
Edited by G. C. Greubel, Dec 25 2025
STATUS
approved
