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A296662
Table read by rows, the odd rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.
3
1, 2, 3, 2, 5, 9, 10, 9, 5, 14, 28, 34, 35, 34, 28, 14, 42, 90, 117, 125, 126, 125, 117, 90, 42, 132, 297, 407, 451, 461, 462, 461, 451, 407, 297, 132, 429, 1001, 1430, 1638, 1703, 1715, 1716, 1715, 1703, 1638, 1430, 1001, 429
OFFSET
0,2
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 diagonal next to the main diagonal of M(2n+1)^(2n+1).
FORMULA
T(n, n) = A001700(n).
T(n, 0) = T(n, 2*n) = A000108(n+1).
T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, n-k-1) for k=0..n.
T(n, k) = binomial(2*n+1, n+1) - binomial(2*n+1, k-n-1) for k=n+1..2*n and n>0.
EXAMPLE
The triangle starts:
0: [ 1]
1: [ 2, 3, 2]
2: [ 5, 9, 10, 9, 5]
3: [ 14, 28, 34, 35, 34, 28, 14]
4: [ 42, 90, 117, 125, 126, 125, 117, 90, 42]
5: [132, 297, 407, 451, 461, 462, 461, 451, 407, 297, 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+1)^(2*n+1), 1), list), n=0..6);
MATHEMATICA
v[n_] := If[n == 1, 1, 0];
m[n_] := MatrixPower[ToeplitzMatrix[Table[v[k], {k, 0, n}]], n];
d[n_] := Diagonal[m[2 n + 1], 1];
Table[d[n], {n, 0, 6}] // Flatten
PROG
(Sage)
def T(n, k):
if k > n:
b = binomial(2*n+1, k - n - 1)
else:
b = binomial(2*n+1, n - k - 1)
return binomial(2*n+1, n+1) - b
for n in (0..6):
print([T(n, k) for k in (0..2*n)])
CROSSREFS
Sequence in context: A293944 A050159 A147294 * A353299 A349790 A335362
KEYWORD
nonn,tabf
AUTHOR
Peter Luschny, Dec 20 2017
STATUS
approved