cp's OEIS Frontend

A008316 Triangle of coefficients of Legendre polynomials P_n (x).

%I A008316 #40 Feb 16 2025 08:32:32
%S A008316 1,1,-1,3,-3,5,3,-30,35,15,-70,63,-5,105,-315,231,-35,315,-693,429,35,
%T A008316 -1260,6930,-12012,6435,315,-4620,18018,-25740,12155,-63,3465,-30030,
%U A008316 90090,-109395,46189,-693,15015,-90090,218790,-230945,88179,231,-18018,225225,-1021020,2078505,-1939938,676039
%N A008316 Triangle of coefficients of Legendre polynomials P_n (x).
%D A008316 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.
%H A008316 T. D. Noe, <a href="/A008316/b008316.txt">Rows n=0..100 of triangle, flattened</a>
%H A008316 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 A008316 T. Copeland, <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera</a>, see the Additional Notes section, 2015.
%H A008316 H. N. Laden, <a href="https://drum.lib.umd.edu/handle/1903/16857">An historical, and critical development of the theory of Legendre polynomials before 1900</a>, Master of Arts Thesis, University of Maryland 1938.
%H A008316 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LegendrePolynomial.html">Legendre Polynomial</a>
%e A008316 Triangle starts:
%e A008316    1;
%e A008316    1;
%e A008316   -1,   3;
%e A008316   -3,   5;
%e A008316    3, -30, 35;
%e A008316   15, -70, 63;
%e A008316   ...
%e A008316 P_5(x) = (15*x - 70*x^3 + 63*x^5)/8 so T(5, ) = (15, -70, 63). P_6(x) = (-5 + 105*x^2 - 315*x^4 + 231*x^6)/16 so T(6, ) = (-5, 105, -315, 231). - _Michael Somos_, Oct 24 2002
%t A008316 Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]]
%o A008316 (PARI) {T(n, k) = if( n<0, 0, polcoeff( pollegendre(n) * 2^valuation( (n\2*2)!, 2), n%2 + 2*k))}; /* _Michael Somos_, Oct 24 2002 */
%Y A008316 Cf. A001790, A001800, A001801.
%Y A008316 With zeros: A100258.
%Y A008316 Cf. A121448.
%K A008316 sign,tabf,easy,nice
%O A008316 0,4
%A A008316 _N. J. A. Sloane_
%E A008316 More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002