A001796 Coefficients of Legendre polynomials. (Formerly M3116 N1263)

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

Offset

0,2

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.

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

Mathematica

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

Prog

(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
(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
(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

Crossrefs

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

Cf. A000108.

Sequence in context: A306442 A080424 A285008 * A174613 A341566 A354655

Adjacent sequences: A001793 A001794 A001795 * A001797 A001798 A001799

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 27 2012

Status

approved