A296663 -id:A296663 - cp's OEIS frontend

A296664 Table read by rows, diagonals of powers of Toeplitz matrices generated by the characteristic function of 1, T(n, k) for n >= 0 and 0 <= k <= 2*floor(n/2).

1, 1, 1, 2, 1, 2, 3, 2, 2, 5, 6, 5, 2, 5, 9, 10, 9, 5, 5, 14, 19, 20, 19, 14, 5, 14, 28, 34, 35, 34, 28, 14, 14, 42, 62, 69, 70, 69, 62, 42, 14, 42, 90, 117, 125, 126, 125, 117, 90, 42, 42, 132, 207, 242, 251, 252, 251, 242, 207, 132, 42

Offset: 0

Peter Luschny, Dec 19 2017

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(n)^n if n is even or the diagonal next to the main diagonal if n is odd. Note that the antidiagonals of M(n)^n are the rows of Pascal's triangle A007318.

			The first few matrices M(n)^n are:
n=0   n=1     n=2       n=3         n=4
|1|  |0 1|  |1 0 1|  |0 2 0 1|  |2 0 3 0 1|
     |1 0|  |0 2 0|  |2 0 3 0|  |0 5 0 4 0|
            |1 0 1|  |0 3 0 2|  |3 0 6 0 3|
                     |1 0 2 0|  |0 4 0 5 0|
                                |1 0 3 0 2|
The triangle starts:
0: [ 1]
1: [ 1]
2: [ 1,  2,   1]
3: [ 2,  3,   2]
4: [ 2,  5,   6,   5,   2]
5: [ 5,  9,  10,   9,   5]
6: [ 5, 14,  19,  20,  19,  14,  5]
7: [14, 28,  34,  35,  34,  28,  14]
8: [14, 42,  62,  69,  70,  69,  62, 42, 14]
9: [42, 90, 117, 125, 126, 125, 117, 90, 42]

Cf. A000108, A001405, A208355, A296663 (row sums), A296662 (odd rows), A296666 (even rows).

v := n -> `if`(n=1, 1, 0):
M := n -> LinearAlgebra:-ToeplitzMatrix([seq(v(j), j=0..n)], symmetric):
seq(convert(ArrayTools:-Diagonal(M(n)^n, n mod 2), list), n=0..10);

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[n], Mod[n, 2]]];
Table[d[n], {n, 0, 10}] // Flatten

def T(n, k):
    h, e = n//2, n%2 == 0
    a = binomial(n, h) if e else binomial(2*h+1, h+1)
    if k > h:
        b = binomial(n, k-h-1) if e else binomial(2*h+1, k-h-1)
    else:
        b = binomial(n, h+k+1) if e else binomial(2*h+1, h-k-1)
    return a - b
for n in (0..9): print([T(n, k) for k in (0..2*(n//2))])

T(n, 0) = T(n, 2*floor(n/2)) = A208355(n) = A000108(floor((n+1)/2)).
T(n, floor(n/2)) = A001405(n).
Further formulas can be found in A296662 and A296666 for the cases n odd and n even.

A296769 Row sums of A296662.

Original entry on oeis.org

1, 7, 38, 187, 874, 3958, 17548, 76627, 330818, 1415650, 6015316, 25413342, 106853668, 447472972, 1867450648, 7770342787, 32248174258, 133530264682, 551793690628, 2276098026922, 9373521044908, 38546133661492, 158301250009768, 649328801880622

Offset: 0

Peter Luschny, Dec 20 2017

nonn

Cf. A000531, A296662, bisection of A296663.

a := n -> 2^(1 + 2*n)*((2*GAMMA(5/2 + n))/(sqrt(Pi)*GAMMA(2 + n)) - 1):
seq(a(n), n=0..23);
# alternative
A296769 := proc(n)
    2^(2*n)*(doublefactorial(2*n+3)/(1+n)/doublefactorial(2*n)-2) ;
end proc:
seq(A296769(n),n=0..10) ; # R. J. Mathar, Jan 03 2018

a(n) = 2^(1 + 2*n)*((2*Gamma(5/2 + n))/(sqrt(Pi)*Gamma(2 + n)) - 1).
a(n) ~ 4^(n+1)*(sqrt(n/Pi) - 1/2).
a(n) = A000531(n+1). - R. J. Mathar, Jan 03 2018
a(n) = A033876(n)-2^(2*n+1). - R. J. Mathar, Jan 03 2018

A296665 Row sums of A296666.

Original entry on oeis.org

1, 4, 20, 96, 444, 2000, 8840, 38528, 166124, 710256, 3016056, 12736064, 53530840, 224107936, 935062544, 3890018816, 16141765964, 66829954736, 276135664664, 1138932645056, 4690042582664, 19285299964256, 79196366286704, 324835930747136, 1330905207444344

Offset: 0

Peter Luschny, Dec 19 2017

nonn

Cf. A296666, bisection of A296663.

a := n -> 4^n*((2*(n + 1)*GAMMA(n + 1/2))/(sqrt(Pi)*GAMMA(n + 1)) - 1);
seq(a(n), n=0..24);

a(n) = 4^n*((2*(n + 1)*Gamma(n + 1/2))/(sqrt(Pi)*Gamma(n + 1)) - 1).

a(n) ~ 4^n*((2*n + 7/4)/sqrt(n*Pi) - 1).

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A296664 Table read by rows, diagonals of powers of Toeplitz matrices generated by the characteristic function of 1, T(n, k) for n >= 0 and 0 <= k <= 2*floor(n/2).

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A296769 Row sums of A296662.

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A296665 Row sums of A296666.

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