A056953 Denominators of continued fraction for alternating factorial.

1, 1, 2, 3, 7, 13, 34, 73, 209, 501, 1546, 4051, 13327, 37633, 130922, 394353, 1441729, 4596553, 17572114, 58941091, 234662231, 824073141, 3405357682, 12470162233, 53334454417, 202976401213, 896324308634, 3535017524403, 16083557845279, 65573803186921

Offset: 0

Author

Aleksandar Petojevic, Sep 05 2000

Keywords

• nonn
• easy

Comments

Starting (1, 2, 3, ...) with offset 0 = eigensequence of an infinite lower triangular matrix with 1's in the main diagonal and the natural numbers repeated in the subdiagonal. - Gary W. Adamson, Feb 14 2011

a(n) is the number of involutions of [n] such that every 2-cycle contains one odd and one even element; a(4) = 7: 1234, 1243, 1324, 2134, 2143, 4231, 4321. - Alois P. Heinz, Feb 14 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, arXiv:2210.17461 [math.RT], 2022.
• Index entries for sequences related to Laguerre polynomials

Crossrefs

Bisections are A000262 and A002720.

Cf. A124428, diagonals of A088699.

Programs

Magma

[(&+[Factorial(k)*Binomial(Floor(n/2),k)*Binomial(Floor((n+1)/2),k): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, May 16 2018

Maple

a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 3][n+1],
      ((4*n-2)*a(n-2) +2*a(n-3) -(n-2)*(n-3)*a(n-4)) /4)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 14 2013

Mathematica

Table[Sum[k!*Binomial[Floor[n/2], k]*Binomial[Floor[(n+1)/2], k] , {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, May 16 2018 *)

PARI

a(n)=sum(k=0,n\2,k!*binomial(n\2,k)*binomial((n+1)\2,k)) \\ Paul D. Hanna, Oct 31 2006

Formula

a(0)=1; a(1)=1; a(n) = a(n-1) + n*a(n-2)/2.

a(n) = Sum_{k=0..[n/2]} k!*C([n/2],k)*C([(n+1)/2],k). - Paul D. Hanna, Oct 31 2006

a(n) ~ n^(n/2 + 1/4) / (2^(n/2 + 3/4) * exp(n/2 - sqrt(2*n) + 1/2)) * (1 + (25 + 6*(-1)^n)/(24*sqrt(2*n)) + (397 + 156*(-1)^n)/(2304*n)). - Vaclav Kotesovec, Feb 22 2019