A008317 Triangle of coefficients of expansions of powers of x in terms of Legendre polynomials P_n(x) over common denominator.

1, 1, 1, 2, 3, 2, 7, 20, 8, 27, 28, 8, 33, 110, 72, 16, 143, 182, 88, 16, 715, 2600, 2160, 832, 128, 3315, 4760, 2992, 960, 128, 4199, 16150, 15504, 7904, 2176, 256, 20349, 31654, 23408, 10080, 2432, 256, 52003, 208012, 220248, 133952, 50048, 10752

Offset

0,4

References

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.

P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.

Links

Table of n, a(n) for n=0..47.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Eric Weisstein's World of Mathematics, Legendre Polynomial

Example

{1}, {1}, {1,2}, {3,2}, {7,20,8}, {27,28,8}, {33,110,72,16}, ...
x^5 = (27*P_1 +28*P_3 + 8*P_5) / 63, so T(5,2) = 8. - Michael Somos, Feb 04 2004

Prog

(PARI) {T(n, m) = my(Q); if( n<0, 0, m = n%2 + m*2; Q = intformal( ^n * pollegendre(m)); (subst(Q, x, 1) - subst(Q, x, -1)) * (2*m+1) / 2 * polcoeff( pollegendre(n), n) * 2^valuation((n\2*2)!, 2))}; /*Michael Somos, Feb 04 2004 */

Crossrefs

A001790 is the common denominator.

Sequence in context: A170842 A014784 A048601 * A139011 A338770 A063708

Adjacent sequences: A008314 A008315 A008316 * A008318 A008319 A008320

Keyword

nonn,tabl

Author

N. J. A. Sloane.

Status

approved