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A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents. 15
1, 0, 1, -1, 0, 3, 0, -3, 0, 5, 3, 0, -30, 0, 35, 0, 15, 0, -70, 0, 63, -5, 0, 105, 0, -315, 0, 231, 0, -35, 0, 315, 0, -693, 0, 429, 35, 0, -1260, 0, 6930, 0, -12012, 0, 6435, 0, 315, 0, -4620, 0, 18018, 0, -25740, 0, 12155, -63, 0, 3465, 0, -30030, 0, 90090, 0, -109395, 0, 46189 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,6
COMMENTS
For a relation to Jacobi quartic elliptic curves, see the MathOverflow link. For a self-convolution of the polynomials relating them to the Chebyshev and Fibonacci polynomials, see A049310 and A053117. For congruences and connections to other polynomials (Jacobi, Gegenbauer, and Chebyshev) see the Allouche et al. link. For relations to elliptic cohomology and modular forms, see references in Copeland link.- Tom Copeland, Feb 04 2016
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
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].
J. Allouche and G. Skordev, Schur congruences, Carlitz sequences of polynomials and automaticity, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, p. 21-49.
H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
FORMULA
The n-th normalized Legendre polynomial is generated by 2^(-n-a(n)) (d/dx)^n (x^2-1)^n / n! with a(n) = A005187(n/2) for n even and a(n) = A005187((n-1)/2) for n odd. The non-normalized polynomials have the o.g.f. 1 / sqrt(1 - 2xz + z^2). - Tom Copeland, Feb 07 2016
The consecutive nonzero entries in the m-th row are, in order, (c+b)!/(c!(m-b)!(2b-m)!*A048896(m-1)) with sign (-1)^b where c = m/2-1, m/2, m/2+1, ..., (m-1) and b = c+1 if m is even and sign (-1)^c with c = (m-1)/2, (m-1)/2+1, (m-1)/2+2, ..., (m-1) with b = c+1 if m is odd. For the 9th row the 5 consecutive nonzero entries are 315, -4620, 18018, -25740, 12155 given by c = 4,5,6,7,8 and b = 5,6,7,8,9. - Richard Turk, Aug 22 2017
EXAMPLE
Triangle begins:
1;
0, 1;
-1, 0, 3;
0, -3, 0, 5;
3, 0, -30, 0, 35;
0, 15, 0, -70, 0, 63;
-5, 0, 105, 0, -315, 0, 231;
0, -35, 0, 315, 0, -693, 0, 429;
35, 0, -1260, 0, 6930, 0, -12012, 0, 6435;
...
MATHEMATICA
row[n_] := CoefficientList[ LegendreP[n, x], x]*2^IntegerExponent[n!, 2]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 15 2015 *)
PROG
(PARI) a(k, n)=polcoeff(pollegendre(k, x), n)*2^valuation(k!, 2)
(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**sum(a007814(i) for i in range(2, n+1))
l=chop(taylor(lambda x: legendre(n, x), 0, n))
print([int(i*y) for i in l]) # Indranil Ghosh, Jul 02 2017
CROSSREFS
Without zeros: A008316. Row sums are A060818.
Columns (with interleaved zeros and signs) include A001790, A001803, A100259. Diagonals include A001790, A001800, A001801, A001802.
Sequence in context: A333791 A323135 A356205 * A045763 A132748 A022901
KEYWORD
sign,tabl
AUTHOR
Ralf Stephan, Nov 13 2004
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)