A001801 Coefficients of Legendre polynomials. (Formerly M3001 N1216)

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

0,1

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].

Michel Bataille and Robert Frontczak, Combinatorial sums derived from properties of Legendre polynomials, arXiv:2604.25981 [math.NT], 2026. See p. 2.

Milan Janjic, Some classes of numbers and derivatives, J. Int. Seq. 12 (2009), Article 09.8.3.

Formula

a(n) = 3*2^(n-3)*binomial(n + 3/2, n)*2^A011371(n+4). - 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 *)

Prog

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

Crossrefs

Cf. A001790, A001796, A008316, A011371.

Bisection of A004733.

Diagonal 3 of triangle A100258.

Sequence in context: A376054 A220747 A088989 * A323551 A267840 A382838

Adjacent sequences: A001798 A001799 A001800 * A001802 A001803 A001804

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos, Oct 25 2002

Status

approved