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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
1, 1, 1, 2, 3, 2, 7, 20, 8, 27, 28, 8, 33, 110, 72, 16, 143, 182, 88, 16, 715, 2600, 2160, 832, 128, 3315, 4760, 2992, 960, 128, 4199, 16150, 15504, 7904, 2176, 256, 20349, 31654, 23408, 10080, 2432, 256, 52003, 208012, 220248, 133952, 50048, 10752 (list; table; graph; refs; listen; history; text; internal format)
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.
P. J. Davis, Interpolation and Approximation, Dover Publications, 1975, p. 372.
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].
Eric Weisstein's World of Mathematics, Legendre Polynomial
EXAMPLE
{1}, {1}, {1,2}, {3,2}, {7,20,8}, {27,28,8}, {33,110,72,16}, ...
x^5 = (27*P_1 +28*P_3 + 8*P_5) / 63, so T(5,2) = 8. - Michael Somos, Feb 04 2004
PROG
(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 */
CROSSREFS
A001790 is the common denominator.
Sequence in context: A170842 A014784 A048601 * A139011 A338770 A063708
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved

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Last modified March 18 22:09 EDT 2024. Contains 370951 sequences. (Running on oeis4.)