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A296666
Table read by rows, the even rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.
4
1, 1, 2, 1, 2, 5, 6, 5, 2, 5, 14, 19, 20, 19, 14, 5, 14, 42, 62, 69, 70, 69, 62, 42, 14, 42, 132, 207, 242, 251, 252, 251, 242, 207, 132, 42, 132, 429, 704, 858, 912, 923, 924, 923, 912, 858, 704, 429, 132
OFFSET
0,3
COMMENTS
Let v be the characteristic function of 1 (A063524) and M(n) for n >= 0 the symmetric Toeplitz matrix generated by the initial segment of v, then row n is the main diagonal of M(2n)^(2n).
Seems to be A050157 + its reflection. - Andrey Zabolotskiy, Dec 19 2017
LINKS
Peter Luschny, Row n for n = 0..30
FORMULA
T(n, 0) = T(n, 2*n) = A000108(n).
T(n, n) are the central binomial coefficients A000984(n).
T(n, k) = binomial(2*n, n) - binomial(2*n, n+k+1) for k=0..n.
T(n, k) = binomial(2*n, n) - binomial(2*n, k-n-1) for k=n+1..2*n and n>0.
EXAMPLE
0: [ 1]
1: [ 1, 2, 1]
2: [ 2, 5, 6, 5, 2]
3: [ 5, 14, 19, 20, 19, 14, 5]
4: [ 14, 42, 62, 69, 70, 69, 62, 42, 14]
5: [ 42, 132, 207, 242, 251, 252, 251, 242, 207, 132, 42]
6: [132, 429, 704, 858, 912, 923, 924, 923, 912, 858, 704, 429, 132]
MAPLE
v := n -> `if`(n=1, 1, 0);
B := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j), j=0..n)], symmetric):
seq(convert(ArrayTools:-Diagonal(B(2*n)^(2*n)), list), n=0..10);
MATHEMATICA
v[n_] := If[n == 1, 1, 0];
m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
d[n_] := If[n == 0, {1}, Diagonal[m[2 n]]];
Table[d[n], {n, 0, 6}] // Flatten
PROG
(Sage)
def T(n, k):
if k > n:
b = binomial(2*n, k - n - 1)
else:
b = binomial(2*n, n + k + 1)
return binomial(2*n, n) - b
for n in (0..6):
print([T(n, k) for k in (0..2*n)])
CROSSREFS
Sequence in context: A019910 A084309 A047991 * A120898 A153910 A208021
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Dec 19 2017
STATUS
approved