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A008316
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Triangle of coefficients of Legendre polynomials P_n (x).
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12
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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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.
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LINKS
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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
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EXAMPLE
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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
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MATHEMATICA
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Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]]
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PROG
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(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 */
(Python)
from mpmath import *
mp.dps=20
def a007814(n): return 1 + bin(n - 1)[2:].count('1') - bin(n)[2:].count('1')
for n in range(11):
y=2**(a007814(int(fac(n))))
l=(chop(taylor(lambda x: legendre(n, x), 0, n)))
print list(filter(lambda i: i!=0, [int(i*y) for i in l])) # Indranil Ghosh, Jul 02 2017
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CROSSREFS
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Cf. A001790, A001800, A001801.
With zeros: A100258.
Cf. A121448.
Sequence in context: A094439 A122037 A201454 * A290284 A258802 A072820
Adjacent sequences: A008313 A008314 A008315 * A008317 A008318 A008319
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KEYWORD
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sign,tabf,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002
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STATUS
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approved
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