A327495 a(n) = numerator( Sum_{j=0..n} (j!/(2^j*floor(j/2)!)^2)^2 ).

1, 17, 69, 1113, 17817, 285297, 1141213, 18260633, 1168681737, 18699007017, 74796032037, 1196736992841, 19147791938817, 306364680039081, 1225458720340365, 19607339566855065, 5019478929156305865, 80311662878468159865, 321246651514020383485, 5139946424277661728785

Offset: 0

Author

Peter Luschny, Sep 27 2019

Keywords

• frac
• nonn

Comments

This sequence is a variant of the Landau constants when the normalized central binomial is replaced by the normalized swinging factorial.

(1) A277233(n)/4^A005187(n) are the Landau constants. These constants are defined as G(n) = Sum_{j=0..n} g(j)^2 with the normalized central binomial

g(n) = (2*n)! / (2^n*n!)^2 = A001790(n)/A046161(n).

(2) A327495(n)/4^A327492(n) are the rationals considered here. These numbers are defined as H(n) = Sum_{j=0..n} h(j)^2 with the normalized swinging factorial

h(n) = n! / (2^n*floor(n/2)!)^2 = A163590(n)/A327493(n).

(3) In particular, this means that we have the pure integer representations

A277233(n) = Sum_{k=0..n}(A001790(k)*(2^(A005187(n) - A005187(k))))^2;

A327495(n) = Sum_{k=0..n}(A163590(k)*(2^(A327492(n) - A327492(k))))^2.

(4) A163590 is the odd part of the swinging factorial and A001790 is the odd part of the swinging factorial at even indices (see a comment from Aug 01 2009 in A001790). Similarly, A327493(2n)=A046161(2n) and A327493(2n+1) = 2*A046161(2n+1).

(5) A005187 are the partial sums of A001511, the 2-adic valuation of 2n, and A327492 are the partial sums of A327491.

Examples

			r(n) = 1, 17/16, 69/64, 1113/1024, 17817/16384, 285297/262144, 1141213/1048576, 18260633/16777216, ...
		

Crossrefs

Denominators in A327496.

Cf. A327491, A327492, A327493, A327494.

Cf. A277233, A005187, A056040, A000984, A001790/A046161, A163590, A001511.

Programs

Maple

A327495 := n -> numer(add(j!^2/(2^j*iquo(j,2)!)^4, j=0..n)):
seq(A327495(n), n=0..19);

PARI

a(n)={ numerator(sum(j=0, n, (j!/(2^j*(j\2)!)^2)^2 )) } \\ Andrew Howroyd, Sep 28 2019

Formula

Denominator(r(n)) = 4^A327492(n) = A327493(n)^2 = A327496(n).

a(n) = Sum_{k=0..n} (A163590(k)*(2^(A327492(n) - A327492(k))))^2.