A001796 Coefficients of Legendre polynomials.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

References

• 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).

Links

• 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.

Crossrefs

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

Cf. A000108.

Programs

Magma

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

Mathematica

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

PARI

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

PARI

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

SageMath

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

Formula

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

Extensions

More terms from Sean A. Irvine, Nov 27 2012