|
|
A136523
|
|
Triangle T(n,k) = A053120(n,k) + A053120(n-1,k), read by rows.
|
|
1
|
|
|
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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,6
|
|
LINKS
|
|
|
FORMULA
|
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).
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
|
A053120[n_, k_]:= Coefficient[ChebyshevT[n, x], x, k];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
|
|
PROG
|
(Magma)
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;
(SageMath)
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)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|