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A008317
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Triangle of coefficients of expansions of powers of x in terms of Legendre polynomials P_n(x) over common denominator.
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0
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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
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OFFSET
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0,4
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REFERENCES
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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.
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LINKS
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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].
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EXAMPLE
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{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
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PROG
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(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 */
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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