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A123023 a(n) = (n-2)*a(n-2), a(0)=1, a(1)=1. 11
1, 1, 0, 1, 0, 3, 0, 15, 0, 105, 0, 945, 0, 10395, 0, 135135, 0, 2027025, 0, 34459425, 0, 654729075, 0, 13749310575, 0, 316234143225, 0, 7905853580625, 0, 213458046676875, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

Old name was: a(n) = b(n)*n!, where b(n+2) = n*b(n)/((n+2)*(n+1)), b(0) = 0, b(1) = 1.

a(n) is the number of ways of separating n terms into pairs. - Stephen Crowley, Apr 07 2007

REFERENCES

Richard Bronson, Schaum's Outline of Modern Introductory Differential Equations, MacGraw-Hill, New York,1973, page 107, solved problem 19.18

Norbert Wiener, Nonlinear Problems in Random Theory, 1958, Equation 1.31

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..800

Michael Wallner, A bijection of plane increasing trees with relaxed binary trees of right height at most one, arXiv:1706.07163 [math.CO], 2017, Table 2 on p. 13.

FORMULA

a(n) = (1/2)*GAMMA((1/2)*n + 1/2)*2^((1/2)*n)*(1 + (-1)^n)/sqrt(Pi). - Stephen Crowley, Apr 07 2007

With offset -1, E.g.f.: exp(x^2/2). - Geoffrey Critzer, Mar 15 2009

a(2n) = A001147(n-1). - R. J. Mathar, Oct 11 2011

From Sergei N. Gladkovskii, Nov 18 2012, Dec 05 2012, May 16 2013, May 24 2013, Jun 07 2013: (Start)

Continued fractions:

E.g.f.: 1 + x*E(0) where E(k) = 1 + x^2*(4*k+1)/((4*k+2)*(4*k+3) - x^2*(4*k+2)*(4*k+3)^2/(x^2*(4*k+3) + (4*k+4)*(4*k+5)/E(k+1))).

G.f.: 1 + x/G(0) where G(k) =  1 - x^2*(k+1)/G(k+1).

G.f.: 1 + x + x^3/(1+x) + Q(0)*x^4/(1+x), where Q(k) = 1 + (2*k+3)*x/(1 - x/(x + 1/Q(k+1))).

G.f.: 1 + x*G(0)/2, where G(k) = 1 + 1/(1-x/(x+1/x/(2*k+1)/G(k+1))).

G.f.: (G(0) - 1)*x^2/(1+x) + 1 + x, where G(k) = 1 + x*(2*k+1)/(1 - x/(x + 1/G(k+1))). (End)

MAPLE

with(combstruct):ZL2:=[S, {S=Set(Cycle(Z, card=2))}, labeled]:seq(count(ZL2, size=n), n=1..28); # Zerinvary Lajos, Sep 24 2007

MATHEMATICA

b[n_] := b[n] = (n - 2) * b[n - 2]/(n * (n - 1)); b[0] = 1; b[1] = 1; Table[b[n] * n!, {n, 0, 30}]

RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == (n - 2) a[n - 2]}, a[n], {n, 0, 31}] (* Ray Chandler, Jul 30 2015 *)

CROSSREFS

Sequence in context: A167339 A277936 A138540 * A130637 A054882 A086479

Adjacent sequences:  A123020 A123021 A123022 * A123024 A123025 A123026

KEYWORD

nonn,easy

AUTHOR

Roger L. Bagula, Sep 24 2006

EXTENSIONS

Edited by N. J. A. Sloane, Jan 06 2008

Better name by Sergei N. Gladkovskii, May 24 2013

STATUS

approved

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Last modified February 18 02:20 EST 2018. Contains 299297 sequences. (Running on oeis4.)