A087208 Expansion of e.g.f.: exp(x)/(1-x^2).

1, 1, 3, 7, 37, 141, 1111, 5923, 62217, 426457, 5599531, 46910271, 739138093, 7318002277, 134523132927, 1536780478171, 32285551902481, 418004290062513, 9879378882159187, 142957467201379447, 3754163975220491061, 60042136224579367741, 1734423756551866870183

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..449

Formula

a(n) = Sum_{k=0..floor(n/2)} n!/(n-2*k)!.

a(n) = n*(n-1)*a(n-2) + 1. - Vladeta Jovovic, Aug 24 2004

a(n) = (A000522(n) + (-1)^n*A000166(n))/2. - Vladeta Jovovic, Aug 24 2004

a(n) = Sum_{k=0..n} binomial(n, k)*(1+(-1)^k)k!/2. Binomial transform of A010050 (with interpolated zeros). - Paul Barry, Sep 14 2004

a(n) = Sum_{k=0..n} P(n, k)[1, 0, 1, 0, 1, 0, ...](k). - Ross La Haye, Aug 29 2005

a(n) = (1/(2*exp(1))) * (Integral_{t=0..2} t^n*exp(1-abs(1-t)) dt + Integral_{t=0..oo} ((2+t)^n + (-t)^n) * exp(-t) dt). - Groux Roland, Jan 15 2011

E.g.f.: 1/U(0) where U(k) = 1 - x^2/(1 - 1/(1 + x*(k+1)/U(k+1)); (continued fraction, 3-step). - Sergei N. Gladkovskii, Oct 16 2012

If n is even then a(n) ~ n!*(e/2 + 1/(2*e)) = 1.543080634815243... * n!, if n is odd then a(n) ~ n!*(e/2 - 1/(2*e)) = 1.175201193643801... * n!. - Vaclav Kotesovec, Nov 20 2012

Conjecture: a(n) -a(n-1) -n*(n-1)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, May 29 2013

From Peter Bala, Sep 05 2022: (Start)

The e.g.f. A(x) satisfies the differential equation (x^2 - 1)*A'(x) + (1 + 2*x - x^2)*A(x) = 0 with A(0) = 1. Mathar's recurrence above follows from this.

For k a positive integer, reducing the sequence modulo k produces a purely periodic sequence whose period divides k. For example, modulo 5 the sequence becomes [1, 1, 3, 2, 2, 1, 1, 3, 2, 2, ...] of period 5. (End)

Mathematica

With[{nn=20}, CoefficientList[Series[Exp[x]/(1-x^2), {x, 0, nn}], x] Range[ 0, nn]!] (* Harvey P. Dale, Aug 11 2017 *)

Crossrefs

Cf. A002747.

Sequence in context: A356668 A057625 A308392 * A161675 A208809 A086031

Adjacent sequences: A087205 A087206 A087207 * A087209 A087210 A087211

Keyword

nonn,easy

Author

Vladeta Jovovic, Oct 19 2003

Extensions

Definition clarified by Harvey P. Dale, Aug 11 2017

Status

approved