OFFSET
0,6
LINKS
G. C. Greubel, Rows n = 0..50 of the triangle, flattened
FORMULA
Sum_{k=0..n} T(n, k) = A040000(n).
From G. C. Greubel, Jul 26 2023: (Start)
T(n, 0) = A057077(n).
T(n, 1) = (-1)^floor((n-1)/2) * A109613(n-1).
T(n, 2) = (-1)^floor((n-2)/2) * A008794(n-1).
T(n, 3) = (-1)^floor((n+1)/2) * A000330(n-1).
T(n, n) = A011782(n).
T(n, n-1) = A011782(n-1).
T(n, n-2) = -A001792(n-2).
T(n, n-4) = A001793(n-3).
T(n, n-6) = -A001794(n-6).
Sum_{k=0..n} (-1)^k*T(n,k) = A000007(n).
Sum_{k=0..floor(n/2)} T(n-k, k) = A000007(n) + [n=1].
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (-1)^floor(n/2)*A025192(floor(n/2)). (End)
EXAMPLE
Triangle begins as:
1;
1, 1;
-1, 1, 2;
-1, -3, 2, 4;
1, -3, -8, 4, 8;
1, 5, -8, -20, 8, 16;
-1, 5, 18, -20, -48, 16, 32;
-1, -7, 18, 56, -48, -112, 32, 64;
1, -7, -32, 56, 160, -112, -256, 64, 128;
1, 9, -32, -120, 160, 432, -256, -576, 128, 256;
-1, 9, 50, -120, -400, 432, 1120, -576, -1280, 256, 512;
MATHEMATICA
PROG
(Magma)
function A053120(n, k)
if ((n+k) mod 2) eq 1 then return 0;
elif n eq 0 then return 1;
else return (-1)^Floor((n-k)/2)*(n/(n+k))*Binomial(Floor((n+k)/2), k)*2^k;
end if;
end function;
[A136523(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 26 2023
(SageMath)
def A053120(n, k):
if (n+k)%2==1: return 0
elif n==0: return 1
else: return floor((-1)^((n-k)//2)*(n/(n+k))*binomial((n+k)//2, k)*2^k)
flatten([[A136523(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 26 2023
CROSSREFS
KEYWORD
AUTHOR
Roger L. Bagula, Mar 23 2008
EXTENSIONS
Edited by G. C. Greubel, Jul 26 2023
STATUS
approved