A001801 Coefficients of Legendre polynomials.

3, 15, 105, 315, 6930, 18018, 90090, 218790, 2078505, 4849845, 22309287, 50702925, 1825305300, 4071834900, 18032411700, 39671305740, 347123925225, 755505013725, 3273855059475, 7064634602025, 121511715154830, 260382246760350, 1112542327066950, 2370198870707850, 20146690401016725

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 798.
• G. Prévost, Tables de Fonctions Sphériques. Gauthier-Villars, Paris, 1933, pp. 156-157.
• 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..1000
• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
• Milan Janjic, Some classes of numbers and derivatives, JIS 12 (2009) 09.8.3.

Crossrefs

Cf. A001790, A001796, A008316, A011371.

Bisection of A004733.

Diagonal 3 of triangle A100258.

Programs

Magma

A001801:= func< n | 3*Binomial(n+3,3)*Catalan(n+2)*2^(Valuation(Factorial(n+4),2)-n-4) >;
[A001801(n): n in [0..30]]; // G. C. Greubel, Apr 26 2025

Mathematica

A001801[n_]:= 3*2^(2*n+1)*Binomial[n+3/2, n]/2^DigitCount[n+4,2,1];
Table[A001801[n], {n,0,40}] (* G. C. Greubel, Apr 26 2025 *)

PARI

a(n)=if(n<0,0,polcoeff(pollegendre(n+4),n)*2^valuation((n\2*2+4)!,2))

SageMath

def A001801(n): return 3*2^(n-3)*binomial(n+3/2,n)*2^valuation(factorial(n+4), 2)
print([A001801(n) for n in range(31)]) # G. C. Greubel, Apr 26 2025

Formula

a(n) = 3*2^(n-3)*binomial(n + 3/2, n)*2^A011371(n+4). - G. C. Greubel, Apr 26 2025

Extensions

More terms from Michael Somos, Oct 25 2002