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A117513 Number of ways of arranging 2n tokens in a row, with 2 copies of each token from 1 through n, such that between every pair of tokens labeled i (i=1..n-1) there is exactly one taken labeled i+1. 4
1, 2, 12, 136, 2480, 66336, 2446528, 118984832, 7378078464, 568142287360, 53189920492544, 5949749335001088, 783686338494312448, 120058889459865165824, 21166245289132322242560 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Contribution from Paul Barry, Oct 12 2009: (Start)

The aerated sequence is a(n)=(2^(n/2-1)+0^(n/2)/2)*((1+(-1)^n)/2)*n!*[x^n](1+x*tan(x/2)).

Multiples of the unsigned Genocchi numbers A110501: (1,1,3,17,155,...)*(1,2,4,8,16,...) (End)

LINKS

Table of n, a(n) for n=1..15.

Guo-Niu Han, Enumeration of Standard Puzzles

Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]

FORMULA

G.f.: 1/(1-2x/(1-4x/(1-8x/(1-12x/(1-18x/(1-24x/(1-32x/(1-.../(1-2*floor((n+2)^2/4)*x/(1-... (continued fraction). [From Paul Barry, Dec 03 2009]

G.f.: T(0), where T(k) = 1 - x*(2*k+2)*(k+1)/( x*(2*k+2)*(k+1) - 1/( 1 - x*(2*k+2)*(k+2)/( x*(2*k+2)*(k+2) - 1/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 24 2013

PROG

(Sage) # Algorithm of L. Seidel (1877)

# n -> [a(1), ..., a(n)] for n >= 1.

def A117513_list(n) :

    D = [0]*(n+2); D[1] = 1

    R = []; z = 1/2; b = True

    for i in(0..2*n-1) :

        h = i//2 + 1

        if b :

            for k in range(h-1, 0, -1) : D[k] += D[k+1]

            z *= 2

        else :

            for k in range(1, h+1, 1) :  D[k] += D[k-1]

        b = not b

        if b : R.append(D[h]*z)

    return R

A117513_list(15) # Peter Luschny, Jun 29 2012

CROSSREFS

Cf. A002105, A117514, A117515.

Sequence in context: A201561 A205886 A108996 * A297078 A185522 A119819

Adjacent sequences:  A117510 A117511 A117512 * A117514 A117515 A117516

KEYWORD

nonn

AUTHOR

Nan Zang (nzang(AT)cs.ucsd.edu), Apr 28 2006

EXTENSIONS

More terms from Paul Barry, Oct 12 2009

STATUS

approved

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Last modified October 22 12:40 EDT 2018. Contains 316446 sequences. (Running on oeis4.)