A124399 a(n) = 4^(n - bitcount(n)) where bitcount(n) = A000120(n).

1, 1, 4, 4, 64, 64, 256, 256, 16384, 16384, 65536, 65536, 1048576, 1048576, 4194304, 4194304, 1073741824, 1073741824, 4294967296, 4294967296, 68719476736, 68719476736, 274877906944, 274877906944, 17592186044416, 17592186044416

Offset

0,3

Comments

Numerators of one half of norm square of monic Legendre polynomials p_n(x).

The denominators of these polynomials are given by A069955.

The rationals N2(n) = 2*a(n)/A069955(n) give the minimal norm square for real monic polynomials. The norm square is defined as integral over the interval [-1,+1] of the square of the polynomials. Cf. the Courant-Hilbert reference.

References

Richard Courant and David Hilbert, Methoden der mathematischen Physik, Bd. I, 3, Auflage, Springer, 1993, pp. 73-74.

Links

Table of n, a(n) for n=0..25.

Wolfdieter Lang, Norm square, rationals and more.

Formula

a(n) = numerator(N2(n)/2) with N2(n)/2:=(1/(2*n+1))*((2^n)/binomial(2*n,n))^2.

N2(n)/2 = (1/(2*n+1))*(1/L(n))^2 with L(n)= A001790(n)/A060818(n), the leading coefficient of the Legendre polynomial P_n(x), in lowest terms.

Bisection: a(2*n)=a(2*n+1) = A056982(n), n>=0.

a(n) = numerator((2^n*(n!)^2/(1+2*n)!)^2). - Stefano Spezia, Jan 17 2026

Example

Rationals a(n)/A069955(n): [1, 1/3, 4/45, 4/175, 64/11025, 64/43659, 256/693693, ...].
Rationals N2(n): [2, 2/3, 8/45, 8/175, 128/11025, 128/43659, 512/693693,...].

Mathematica

a[n_] := 4^(n - DigitCount[n, 2, 1]); Array[a, 25, 0] (* Amiram Eldar, Jul 25 2023 *)

Prog

(PARI) a(n) = numerator((1/(2*n+1))*((2^n)/binomial(2*n, n))^2); \\ Michel Marcus, Aug 11 2019
(Julia)
bitcount(n) = sum(digits(n, base=2))
a(n) = 4^(n - bitcount(n)) # Peter Luschny, Oct 01 2019

Crossrefs

Cf. A000120, A001790, A056982, A060818, A069955 (denominators of N2(n) as defined in the comments).

Sequence in context: A212328 A214615 A206489 * A119600 A244027 A219796

Adjacent sequences: A124396 A124397 A124398 * A124400 A124401 A124402

Keyword

nonn,easy

Author

Wolfdieter Lang, Nov 10 2006

Extensions

New name by Peter Luschny, Oct 01 2019

Status

approved