A356205 - OEIS

%I #5 Jul 29 2022 14:14:36

%S 1,0,1,-1,0,3,0,-3,0,5,3,0,-15,0,35,0,15,0,-35,0,63,-5,0,105,0,-315,0,

%T 231,0,-35,0,315,0,-693,0,429,35,0,-315,0,3465,0,-3003,0,6435,0,315,0,

%U -1155,0,9009,0,-6435,0,12155,-63,0,3465,0,-15015,0,45045,0,-109395,0,46189

%N T(n,k) are the numerators of the coefficients of the Legendre polynomials of degree n, with increasing exponents, where T(n,k) is a triangle read by rows.

%e The triangle begins:

%e    1;

%e    0,   1;

%e   -1,   0,    3;

%e    0,  -3,    0,     5;

%e    3,   0,  -15,     0,   35;

%e    0,  15,    0,   -35,    0,   63;

%e   -5,   0,  105,     0, -315,    0,   231;

%e    0, -35,    0,   315,    0, -693,     0,   429;

%e   35,   0, -315,     0, 3465,    0, -3003,     0, 6435;

%e    0, 315,    0, -1155,    0, 9009,     0, -6435,    0, 12155

%e .

%e Fractions:

%e    \ k 0        1       2      3       4       5      6       7      8

%e   n \ -------------------------------------------------------------------

%e   0 |  1        .       .      .       .       .       .      .      .

%e   1 |  0       1        .      .       .       .       .      .      .

%e   2 | -1/2     0       3/2     .       .       .       .      .      .

%e   3 |  0      -3/2     0      5/2      .       .       .      .      .

%e   4 |  3/8     0     -15/4    0      35/8      .       .      .      .

%e   5 |  0      15/8     0    -35/4     0      63/8      .      .      .

%e   6 | -5/16    0     105/16   0    -315/16    0     231/16    .      .

%e   7 |  0     -35/16    0    315/16    0    -693/16    0    429/16    .

%e   8 | 35/128   0    -315/32   0    3465/64    0   -3003/32   0   6435/128

%o (PARI) for (n=0, 10, my(P=pollegendre(n,'x));for (j=0, n, print1(numerator(polcoef(P,j)),", ")); print())

%Y A356206 are the corresponding denominators.

%Y Cf. A005187, A100258.

%K sign,tabl,frac

%O 0,6

%A _Hugo Pfoertner_, Jul 29 2022