A001802 -id:A001802 - cp's OEIS frontend

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

Ralf Stephan, Nov 13 2004

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

			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;
  ...

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.

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.

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.

row[n_] := CoefficientList[ LegendreP[n, x], x]*2^IntegerExponent[n!, 2]; Table[row[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 15 2015 *)

a(k,n)=polcoeff(pollegendre(k,x),n)*2^valuation(k!,2)

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

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

A001796 Coefficients of Legendre polynomials.

Original entry on oeis.org

1, 3, 27, 143, 3315, 20349, 260015, 1710855, 92116035, 631165425, 8775943605, 61750730457, 1755702867191, 12587970424449, 181858466731095, 1322239639929719, 154702037871777123, 1137023085979691001, 16789716964765636633

Offset: 0

N. J. A. Sloane

nonn

Numerators in expansion of c(x)^(3/2), c(x) the g.f. of A000108. - Gerald McGarvey, Oct 07 2008

Coefficient of Legendre_1(x) when x^n is written in term of Legendre polynomials. - Michel Marcus, May 28 2013

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..830
H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.

Cf. A001795, A001797, A001798, A001799, A001800, A001801, A001802.

Cf. A000108.

A001796:= func< n | Numerator(3*(n+1)*Catalan(2*n+1)/(4^n*(2*n+3))) >;
[A001796(n): n in [0..25]]; // G. C. Greubel, Apr 23 2025

Table[Numerator[3*Binomial[2*n+1/2, n]/(2*n+3)], {n,0,30}] (* G. C. Greubel, Apr 23 2025 *)

my(x='x+O('x^30)); apply(numerator, Vec(((1-sqrt(1-4*x))/(2*x))^(3/2))) \\ Michel Marcus, Feb 04 2022

a(n)=numerator(3*binomial(2*n+1/2, n)/(2*n+3)) \\ Tani Akinari, Oct 31 2024

def A001796(n): return numerator(3*binomial(2*n+1/2, n)/(2*n+3))
print([A001796(n) for n in range(31)]) # G. C. Greubel, Apr 23 2025

Numerators of g.f. ((1-sqrt(1-4*x))/(2*x))^(3/2). - Sean A. Irvine, Nov 27 2012

a(n) = numerator(3*binomial(2*n+1/2, n)/(2*n+3)). - Tani Akinari, Oct 31 2024

More terms from Sean A. Irvine, Nov 27 2012

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

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A001796 Coefficients of Legendre polynomials.

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