cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Peter Luschny, Dec 19 2017

Keywords

Crossrefs

Cf. A296666, bisection of A296663.

Programs

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

Formula

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).

A296662 Table read by rows, the odd rows of A296664, T(n, k) for n >= 0 and 0 <= k <= 2n.

Original entry on oeis.org

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

Views

Author

Peter Luschny, Dec 20 2017

Keywords

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).

Examples

			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]

Crossrefs

Cf. A000108, A001700, A050158, A296664, A296666, A296769 (row sums).

Programs

  • 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
  • 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)])

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.

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).

Original entry on oeis.org

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

Views

Author

Peter Luschny, Dec 19 2017

Keywords

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(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.

Examples

			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]

Crossrefs

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

Programs

  • Maple
    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);
  • 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[n], Mod[n, 2]]];
Table[d[n], {n, 0, 10}] // Flatten
  • Sage
    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))])

Formula

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.
Showing 1-3 of 3 results.

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