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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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Table of n, a(n) for n=0..47.
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
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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 = (27P_1+28P_3+8P_5)/63, so T(5,2)=8.
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PROG
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(PARI) T(n, m)=local(Q); if(n<0, 0, m=n%2+m*2; Q=intformal(x^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))
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CROSSREFS
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A001790 is common denominator.
Sequence in context: A170842 A014784 A048601 * A139011 A063708 A096488
Adjacent sequences: A008314 A008315 A008316 * A008318 A008319 A008320
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KEYWORD
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nonn,tabl
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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