A001799 Coefficients of Legendre polynomials.

8, 72, 2160, 15504, 220248, 1564920, 89324640, 640807200, 9246847896, 67087213336, 1957095947664, 14342471475696, 211153052281080, 1560676296310488, 185256494416099008, 1379131680653181504, 20598677144877854232

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Coefficient of Legendre_4(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 = 2..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;
A001799:= func< n | 144*B(n+2, 4)*Numerator(B(4*n, 2*n)/2^(4*n))/(5*B(2*n+5, 5)) >;
[A001799(n): n in [2..30]]; // G. C. Greubel, Apr 24 2025

Maple

a:=n->(9*(2*n)*(2*n-2)/((2*n+1)*(2*n+3)*(2*n+5)))*numer(binomial(4*n,2*n)/2^(4*n)); # Sean A. Irvine, Nov 28 2012

Mathematica

A001799[n_]:= With[{B=Binomial}, 144*B[n+2,4]*Numerator[B[4*n,2*n]/2^(4*n)]/(5*B[2*n+ 5, 5])];
Table[A001799[n], {n,2,35}] (* G. C. Greubel, Apr 24 2025 *)

SageMath

b=binomial
def A001799(n): return 144*b(n+2, 4)*numerator(b(4*n, 2*n)/2^(4*n))//(5*b(2*n+5, 5))
print([A001799(n) for n in range(2, 31)]) # G. C. Greubel, Apr 24 2025

Formula

a(n) = (9*(2*n)*(2*n-2)/((2*n+1)*(2*n+3)*(2*n+5))) * numerator(binomial(4*n, 2*n)/2^(4*n)). - Sean A. Irvine, Nov 28 2012

Extensions

More terms from Sean A. Irvine, Nov 28 2012