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A008316
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Triangle of coefficients of Legendre polynomials P_n (x).
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9
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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.
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 * A258802 A072820 A204004
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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