A001795 Coefficients of Legendre polynomials.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Divisor of A048990 and A065097.

Apparently a bisection of A002596.

Bisection of A099024.

Cf. A000108, A001791.

Programs

Magma

A001795:= func< n | Numerator(Catalan(2*n)/4^n) >;
[A001795(n): n in [0..25]]; // G. C. Greubel, Apr 22 2025

Mathematica

Table[Numerator[CatalanNumber[2*n]/4^n], {n,0,30}] (* G. C. Greubel, Apr 22 2025 *)

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

SageMath

def A001795(n): return numerator(catalan_number(2*n)/4^n)
print([A001795(n) for n in range(31)]) # G. C. Greubel, Apr 22 2025

Formula

1/(sqrt(1-x) + sqrt(1+x)) = Sum_{n>=0} (a(n)/b(n))*x^(2*n) 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

a(n) = numerator( A000108(2*n)/4^n ). - G. C. Greubel, Apr 22 2025

Extensions

More terms from Benoit Cloitre, Mar 12 2002