A356205 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.

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, 231, 0, -35, 0, 315, 0, -693, 0, 429, 35, 0, -315, 0, 3465, 0, -3003, 0, 6435, 0, 315, 0, -1155, 0, 9009, 0, -6435, 0, 12155, -63, 0, 3465, 0, -15015, 0, 45045, 0, -109395, 0, 46189

Offset: 0

Author

Hugo Pfoertner, Jul 29 2022

Keywords

• sign
• tabl
• frac

Examples

			The triangle begins:
   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,   231;
   0, -35,    0,   315,    0, -693,     0,   429;
  35,   0, -315,     0, 3465,    0, -3003,     0, 6435;
   0, 315,    0, -1155,    0, 9009,     0, -6435,    0, 12155
.
Fractions:
   \ k 0        1       2      3       4       5      6       7      8
  n \ -------------------------------------------------------------------
  0 |  1        .       .      .       .       .       .      .      .
  1 |  0       1        .      .       .       .       .      .      .
  2 | -1/2     0       3/2     .       .       .       .      .      .
  3 |  0      -3/2     0      5/2      .       .       .      .      .
  4 |  3/8     0     -15/4    0      35/8      .       .      .      .
  5 |  0      15/8     0    -35/4     0      63/8      .      .      .
  6 | -5/16    0     105/16   0    -315/16    0     231/16    .      .
  7 |  0     -35/16    0    315/16    0    -693/16    0    429/16    .
  8 | 35/128   0    -315/32   0    3465/64    0   -3003/32   0   6435/128
		

Crossrefs

A356206 are the corresponding denominators.

Cf. A005187, A100258.

Programs

PARI

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