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

1, 1, -1, 3, -3, 5, 3, -30, 35, 15, -70, 63, -5, 105, -315, 231, -35, 315, -693, 429, 35, -1260, 6930, -12012, 6435, 315, -4620, 18018, -25740, 12155, -63, 3465, -30030, 90090, -109395, 46189, -693, 15015, -90090, 218790, -230945, 88179, 231, -18018, 225225, -1021020, 2078505, -1939938, 676039

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.

Links

T. D. Noe, Rows n=0..100 of triangle, flattened

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].

T. Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera, see the Additional Notes section, 2015.

H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.

Eric Weisstein's World of Mathematics, Legendre Polynomial

Example

Triangle starts:
   1;
   1;
  -1,   3;
  -3,   5;
   3, -30, 35;
  15, -70, 63;
  ...
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

Mathematica

Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]]

Prog

(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 */

Crossrefs

Cf. A001790, A001800, A001801.

With zeros: A100258.

Cf. A121448.

Sequence in context: A122037 A363796 A201454 * A335952 A290284 A258802

Adjacent sequences: A008313 A008314 A008315 * A008317 A008318 A008319

Keyword

sign,tabf,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002

Status

approved