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

%I A100258 #46 Mar 17 2021 04:20:21
%S A100258 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,
%T A100258 231,0,-35,0,315,0,-693,0,429,35,0,-1260,0,6930,0,-12012,0,6435,0,315,
%U A100258 0,-4620,0,18018,0,-25740,0,12155,-63,0,3465,0,-30030,0,90090,0,-109395,0,46189
%N A100258 Triangle of coefficients of normalized Legendre polynomials, with increasing exponents.
%C A100258 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
%D A100258 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.
%H A100258 T. D. Noe, <a href="/A100258/b100258.txt">Rows n=0..100 of triangle, flattened</a>
%H A100258 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H A100258 J. Allouche and G. Skordev, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00195-8">Schur congruences, Carlitz sequences of polynomials and automaticity</a>, Discrete Mathematics, Vol. 214, Issue 1-3, 21 March 2000, p. 21-49.
%H A100258 Tom Copeland, <a href="https://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">The Elliptic Lie Triad: Riccati and KdV Equations, Infinigens, and Elliptic Genera</a>
%H A100258 H. N. Laden, <a href="https://drum.lib.umd.edu/handle/1903/16857">An historical, and critical development of the theory of Legendre polynomials before 1900</a>, Master of Arts Thesis, University of Maryland 1938.
%H A100258 Shi-Mei Ma, <a href="http://arxiv.org/abs/1304.6654">On gamma-vectors and the derivatives of the tangent and secant functions</a>, arXiv:1304.6654 [math.CO], 2013.
%H A100258 MathOverflow, <a href="http://mathoverflow.net/questions/82597/geometric-picture-of-invariant-differential-of-an-elliptic-curve">Geometric picture of invariant differential of an elliptic curve</a>, Dec 4 2011.
%F A100258 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
%F A100258 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
%e A100258 Triangle begins:
%e A100258    1;
%e A100258    0,   1;
%e A100258   -1,   0,     3;
%e A100258    0,  -3,     0,   5;
%e A100258    3,   0,   -30,   0,   35;
%e A100258    0,  15,     0, -70,    0,   63;
%e A100258   -5,   0,   105,   0, -315,    0,    231;
%e A100258    0, -35,     0, 315,    0, -693,      0, 429;
%e A100258   35,   0, -1260,   0, 6930,    0, -12012,   0, 6435;
%e A100258   ...
%t A100258 row[n_] := CoefficientList[ LegendreP[n, x], x]*2^IntegerExponent[n!, 2]; Table[row[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Jan 15 2015 *)
%o A100258 (PARI) a(k,n)=polcoeff(pollegendre(k,x),n)*2^valuation(k!,2)
%o A100258 (Python)
%o A100258 from mpmath import *
%o A100258 mp.dps=20
%o A100258 def a007814(n):
%o A100258     return 1 + bin(n - 1)[2:].count('1') - bin(n)[2:].count('1')
%o A100258 for n in range(11):
%o A100258     y=2**sum(a007814(i) for i in range(2, n+1))
%o A100258     l=chop(taylor(lambda x: legendre(n, x), 0, n))
%o A100258     print([int(i*y) for i in l]) # _Indranil Ghosh_, Jul 02 2017
%Y A100258 Without zeros: A008316. Row sums are A060818.
%Y A100258 Columns (with interleaved zeros and signs) include A001790, A001803, A100259. Diagonals include A001790, A001800, A001801, A001802.
%Y A100258 Cf. A049310, A005187, A049600, A053117.
%K A100258 sign,tabl
%O A100258 0,6
%A A100258 _Ralf Stephan_, Nov 13 2004