A167589 The third column of the ED4 array A167584.

1, 10, 93, 1020, 13269, 198990, 3383145, 64276920, 1349846505, 31046064210, 776157686325, 20956154152500, 607730434609725, 18839602224969750, 621707822126431425, 21759750056864358000, 805111392478121276625

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..150

Formula

a(n) = (1/8)*(-1)^(n)*(2*n-5)!!*((4*n^3-11*n)+(16*n^4-40*n^2+9)*(Sum_{k=0..(n-1)} ( (-1)^(k+n)/(2*k+1) ) ).

From Peter Bala, Nov 01 2016: (Start)

a(n) = 3*(2*n + 3)!! * Sum_{k = 0..n-1} (-1)^k/((2*k - 3)*(2*k - 1)*(2*k + 1)*(2*k + 3)*(2*k + 5)).

a(n) ~ Pi*2^(n - 5/2)*((n + 2)/e)^(n + 2).

E.g.f.: (6*arcsin(2*x) + 4*x*sqrt(1 - 4*x^2)*(5 - 8*x^2))/(32*(1 - 2*x)^(5/2)).

a(n) = 10*a(n) + (2*n - 7)*(2*n + 1)*a(n-2) with a(0) = 0, a(1) = 1.

The sequence b(n) := (2*n + 3)!! = (2*n + 4)!/((n + 2)!*2^(n+2)) = A001147(n+2) satisfies the same recurrence with b(0) = 3 and b(1) = 15. This leads to the continued fraction representation a(n) = 1/3*b(n)*( 1/(5 - 15/(10 - 7/(10 + 9/(10 + 33/(10 + ... + (2*n - 7)*(2*n + 1)/(10)))))) ) for n >= 2.

As n -> infinity, 3*a(n)/(A001147(n+2)) -> 9/4!*Pi/4 = 1/(5 - 15/(10 - 7/(10 + 9/(10 + 33/(10 + ... + (2*n - 7)*(2*n + 1)/(10 + ...)))))). (End)

Mathematica

Table[(1/8)*(-1)^(n)*(2*n - 5)!!*((4*n^3 - 11*n) + (16*n^4 - 40*n^2 + 9)*(Sum[(-1)^(k + n)/(2*k + 1), {k, 0, n - 1}])), {n, 1, 50}] (* G. C. Greubel, Jun 17 2016 *)

Crossrefs

Equals the third column of the ED4 array A167584.

Other columns are A024199 and A167588.

Cf. A007509 and A025547 (the sum((-1)^(k+n)/(2*k+1), k=0..n-1) factor), A001147.

Sequence in context: A190989 A224696 A099295 * A192899 A192900 A192901

Adjacent sequences: A167586 A167587 A167588 * A167590 A167591 A167592

Keyword

nonn,easy

Author

Johannes W. Meijer, Nov 10 2009

Status

approved