A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents.

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

Offset: 0

Author

Ralf Stephan, Nov 13 2004

Keywords

• sign
• tabl

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

Examples

			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;
  ...
		

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].
• 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.
• Tom Copeland, The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera
• H. N. Laden, An historical, and critical development of the theory of Legendre polynomials before 1900, Master of Arts Thesis, University of Maryland 1938.
• Shi-Mei Ma, On gamma-vectors and the derivatives of the tangent and secant functions, arXiv:1304.6654 [math.CO], 2013.
• MathOverflow, Geometric picture of invariant differential of an elliptic curve, Dec 4 2011.

Crossrefs

Without zeros: A008316. Row sums are A060818.

Columns (with interleaved zeros and signs) include A001790, A001803, A100259. Diagonals include A001790, A001800, A001801, A001802.

Cf. A049310, A005187, A049600, A053117.

Programs

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 *)

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

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