A077070 Triangle read by rows: T(n,k) is the power of 2 in denominator of coefficients of Legendre polynomials, where n >= 0 and 0 <= k <= n.

0, 1, 1, 3, 2, 3, 4, 4, 4, 4, 7, 5, 6, 5, 7, 8, 8, 7, 7, 8, 8, 10, 9, 10, 8, 10, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 15, 12, 13, 12, 14, 12, 13, 12, 15, 16, 16, 14, 14, 15, 15, 14, 14, 16, 16, 18, 17, 18, 15, 17, 16, 17, 15, 18, 17, 18, 19, 19, 19, 19, 18, 18, 18, 18, 19, 19, 19, 19, 22, 20, 21, 20, 22, 19, 20, 19, 22, 20, 21, 20, 22

Offset

0,4

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

T(n, k) = A007814(A144816(n, k)). - Michel Marcus, Jan 29 2022

T(n, k) = 2*n - wt(n-k) - wt(k) where wt = A000120 is the binary weight. - Kevin Ryde, Jan 29 2022

Example

Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
   0;
   1,  1;
   3,  2,  3;
   4,  4,  4,  4;
   7,  5,  6,  5,  7;
   8,  8,  7,  7,  8,  8;
  10,  9, 10,  8, 10,  9, 10;
  ...

Maple

T:= n-> (p-> seq(padic[ordp](denom(coeff(p, x, i)), 2)
             , i=0..2*n, 2))(orthopoly[P](2*n, x)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jan 25 2022

Mathematica

T[n_, k_] := IntegerExponent[Denominator[Coefficient[LegendreP[2n, x], x, 2k]], 2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2017 *)

Prog

(PARI) {T(n, k) = if( k<0 || k>n, 0, -valuation( polcoeff( pollegendre(2*n), 2*k), 2))}
(PARI) T(n, k) = 2*n - hammingweight(n-k) - hammingweight(k); \\ Kevin Ryde, Jan 29 2022

Crossrefs

Cf. A005187 (column k=0), A101925 (column k=1), A077071 (row sums), A144816 (denominators).

Cf. A000120, A007814.

Sequence in context: A272886 A098822 A131597 * A374560 A075988 A029150

Adjacent sequences: A077067 A077068 A077069 * A077071 A077072 A077073

Keyword

nonn,tabl

Author

Michael Somos, Oct 25 2002

Status

approved