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A008316 Triangle of coefficients of Legendre polynomials P_n (x). 9
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)
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.

LINKS

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

EXAMPLE

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

MATHEMATICA

Flatten[Table[(LegendreP[i, x]/.{Plus->List, x->1})Max[ Denominator[LegendreP[i, x]/.{Plus->List, x->1}]], {i, 0, 12}]]

PROG

(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

CROSSREFS

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

KEYWORD

sign,tabf,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Vit Planocka (planocka(AT)mistral.cz), Sep 28 2002

STATUS

approved

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Last modified August 17 23:58 EDT 2017. Contains 290682 sequences.