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
Examples
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
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
Programs
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Mathematica
Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]] -
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 */
Extensions
More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002