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A008317 Triangle of coefficients of expansions of powers of x in terms of Legendre polynomials P_n(x) over common denominator. 0

%I

%S 1,1,1,2,3,2,7,20,8,27,28,8,33,110,72,16,143,182,88,16,715,2600,2160,

%T 832,128,3315,4760,2992,960,128,4199,16150,15504,7904,2176,256,20349,

%U 31654,23408,10080,2432,256,52003,208012,220248,133952,50048,10752

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

%D 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.

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

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>

%e {1}, {1}, {1,2}, {3,2}, {7,20,8}, {27,28,8}, {33,110,72,16}, ...

%e x^5 = (27*P_1 +28*P_3 + 8*P_5) / 63, so T(5,2) = 8. - _Michael Somos_, Feb 04 2004

%o (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 */

%Y A001790 is the common denominator.

%K nonn,tabl

%O 0,4

%A _N. J. A. Sloane_.

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)