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A008316
Triangle of coefficients of Legendre polynomials P_n (x).
29
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
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].
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
With zeros: A100258.
Cf. A121448.
Sequence in context: A122037 A363796 A201454 * A335952 A290284 A258802
KEYWORD
sign,tabf,easy,nice
EXTENSIONS
More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002
STATUS
approved