A001795 Coefficients of Legendre polynomials. (Formerly M4407 N1861)

1, 1, 7, 33, 715, 4199, 52003, 334305, 17678835, 119409675, 1641030105, 11435320455, 322476036831, 2295919134019, 32968493968795, 238436656380769, 27767032438524099, 203236010537432691, 2989949596465113373

Offset

0,3

Comments

Numerators in expansion of sqrt(c(x)), c(x) the g.f. of A000108. - Paul Barry, Jul 12 2005

Coefficient of Legendre_0(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

T. D. Noe, Table of n, a(n) for n = 0..100

H. E. Salzer, Coefficients for expressing the first twenty-four powers in terms of the Legendre polynomials, Math. Comp., 3 (1948), 16-18.

Formula

1/(sqrt(1-x) + sqrt(1+x)) = Sum_{n>=0} (a(n)/b(n))*x^(2n) where b(n) is a power of 2. - Benoit Cloitre, Mar 12 2002

For n >= 1, 2^(n+1)*a(2^(n-1)) = A001791(2^n). - Vladimir Shevelev, Sep 05 2010

a(n) = numerator(binomial(2*n-1/2, n)/(2*n+1)). - Tani Akinari, Oct 22 2024

Prog

(PARI) my(x='x+O('x^30)); apply(numerator, Vec(((1-sqrt(1-4*x))/(2*x))^(1/2))) \\ Michel Marcus, Feb 04 2022
(PARI) a(n)=numerator(binomial(2*n-1/2, n)/(2*n+1)) \\ Tani Akinari, Oct 22 2024

Crossrefs

Divisor of A048990 and A065097. Apparently a bisection of A002596.

Bisection of A099024.

Cf. A000108, A001791.

Sequence in context: A202762 A202757 A266018 * A336277 A209897 A209814

Adjacent sequences: A001792 A001793 A001794 * A001796 A001797 A001798

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Benoit Cloitre, Mar 12 2002

Status

approved