A001798 Coefficients of Legendre polynomials.

2, 28, 182, 4760, 31654, 428260, 2941470, 163761840, 1152562950, 16381761396, 117402623338, 3390322778024, 24634522766126, 360043025043380, 2644479279859438, 312191499849352032, 2312918756095439814, 34398444513178377492

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Coefficient of Legendre_3(x) when x^n is written in term of Legendre polynomials. - Sean A. Irvine, Nov 28 2012

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 = 1..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. A001796.

Programs

Magma

B:=Binomial;
A001798:= func< n | 14*B(n+2,3)*Numerator(B(4*n+2,2*n+1)/2^(4*n))/B(2*n+5,4) >;
[A001798(n): n in [1..30]]; // G. C. Greubel, Apr 23 2025

Maple

a:=n->(14*n/((2*n+3)*(2*n+5)))*numer(binomial(4*n+2,2*n+1)/2^(4*n)); # Sean A. Irvine, Nov 28 2012

Mathematica

A001798[n_]:= With[{B=Binomial}, 14*B[n+2,3]*Numerator[B[4*n+2,2*n+1]/2^(4*n) ]/B[2*n+5,4]];
Table[A001798[n], {n,30}] (* G. C. Greubel, Apr 23 2025 *)

SageMath

b=binomial
def A001798(n): return 14*b(n+2,3)*numerator(b(4*n+2,2*n+1)/2^(4*n) )//b(2*n+5,4)
print([A001798(n) for n in range(1,31)]) # G. C. Greubel, Apr 23 2025

Formula

a(n) = (14*n/((2*n+3)*(2*n+5)))*numerator(binomial(4*n+2, 2*n+1)/2^(4*n)). - Sean A. Irvine, Nov 28 2012

Extensions

More terms from Sean A. Irvine, Nov 28 2012