A001800 Coefficients of Legendre polynomials.

1, 3, 30, 70, 315, 693, 12012, 25740, 109395, 230945, 1939938, 4056234, 16900975, 35102025, 1163381400, 2404321560, 9917826435, 20419054425, 167890003050, 344616322050, 1412926920405, 2893136075115, 47342226683700, 96742811049300, 395033145117975

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

• Alois P. Heinz, Table of n, a(n) for n = 0..500
• 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].
• Eric Weisstein's World of Mathematics, Legendre Polynomial, eq. 28.

Crossrefs

Cf. A001790, A001796, A001803, A008316.

Diagonal 2 of triangle A100258.

Programs

Magma

A001800:= func< n | (n+1)*(n+2)*Catalan(n+1)/2^(&+Intseq(n+2, 2)) >;
[A001800(n): n in [0..30]]; // G. C. Greubel, Apr 25 2025

Maple

wt:= proc(n) local m, r; m:=n; r:=0;
       while m>0 do r:= r+irem(m, 2, 'm') od; r
     end:
a:= n-> (n+1) *binomial(2*n+2, n+1)/2^wt(n+2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 29 2013

Mathematica

a[n_] := (n+1)*Binomial[2*n+2, n+1]/2^DigitCount[n+2, 2, 1]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Mar 13 2014 *)

PARI

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

SageMath

def A001800(n): return (n+1)*binomial(2*n+2,n+1)//2^sum((n+2).digits(2))
print([A001800(n) for n in range(31)]) # G. C. Greubel, Apr 25 2025

Formula

a(n) = (n+1) * C(2n+2, n+1) / 2^A000120(n+2).

Extensions

More terms from Michael Somos, Oct 25 2002