A004733 -id:A004733 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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).

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.

Cf. A001790, A001796, A008316, A011371.
Bisection of A004733.
Diagonal 3 of triangle A100258.

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

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 *)

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

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

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

More terms from Michael Somos, Oct 25 2002

A004732 Numerator of n!!/(n+3)!!.

Original entry on oeis.org

1, 1, 2, 1, 8, 5, 16, 7, 128, 21, 256, 33, 1024, 429, 2048, 715, 32768, 2431, 65536, 4199, 262144, 29393, 524288, 52003, 4194304, 185725, 8388608, 334305, 33554432, 9694845, 67108864, 17678835, 2147483648

Offset: 0

N. J. A. Sloane

S. Janson, On the traveling fly problem, Graph Theory Notes of New York, Vol. XXXI, 17, 1996.

Robert Israel, Table of n, a(n) for n = 0..3327
S. Janson, On the traveling fly problem

Cf. A000108, A000265, A004733, A007814, A048881.

f:= proc(n) local m,r;
  m:= floor(n/2);
  if n::even then 2^(2*m - padic:-ordp(binomial(2*m+1,m),2) - padic:-ordp(m+1,2))
  else
    r:= binomial(2*m+2,m+1)/(m+2);
    r/2^padic:-ordp(r,2);
fi
end proc:
map(f, [$0..50]); # Robert Israel, Jan 07 2019

Numerator[Table[n!!/(n+3)!!,{n,0,40}]] (* Harvey P. Dale, Nov 26 2015 *)

a(n) = numerator(prod(i=0, floor((n-1)/2), n-2*i)/prod(i=0, floor((n+2)/2), n+3-2*i)) \\ Michel Marcus, May 24 2013

From Robert Israel, Jan 07 2019: (Start)
a(2*m) = 2^(2*m+1 - A048881(m) - A007814(m+1)).
a(2*m+1) = A000265(A000108(m+1)). (End)

cp's OEIS Frontend

A001801 Coefficients of Legendre polynomials.

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