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A056953 Denominators of continued fraction for alternating factorial. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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
Francesca Aicardi, Diego Arcis, and Jesús Juyumaya, Ramified inverse and planar monoids, arXiv:2210.17461 [math.RT], 2022.
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
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 *)
PROG
(PARI) a(n)=sum(k=0, n\2, k!*binomial(n\2, k)*binomial((n+1)\2, k)) \\ Paul D. Hanna, Oct 31 2006
(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
CROSSREFS
Bisections are A000262 and A002720.
Cf. A124428, diagonals of A088699.
Sequence in context: A237255 A129859 A280765 * A321681 A045611 A006840
KEYWORD
nonn,easy
AUTHOR
STATUS
approved

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Last modified February 27 21:03 EST 2024. Contains 370378 sequences. (Running on oeis4.)