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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
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
0,6
EXAMPLE
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
PROG
(PARI) for (n=0, 10, my(P=pollegendre(n, 'x)); for (j=0, n, print1(numerator(polcoef(P, j)), ", ")); print())
CROSSREFS
A356206 are the corresponding denominators.
Sequence in context: A272727 A333791 A323135 * A100258 A045763 A132748
KEYWORD
sign,tabl,frac
AUTHOR
Hugo Pfoertner, Jul 29 2022
STATUS
approved