A001800 Coefficients of Legendre polynomials. (Formerly M3123 N1266)

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

0,2

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.

Formula

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

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 *)

Prog

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

Crossrefs

Cf. A001790, A001801, A008316.

Cf. A001803.

Diagonal 2 of triangle A100258.

Sequence in context: A132084 A208880 A012009 * A152767 A195029 A211617

Adjacent sequences: A001797 A001798 A001799 * A001801 A001802 A001803

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Michael Somos, Oct 25 2002

Status

approved