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A000271 Sums of ménage numbers.
(Formerly M3020 N1222)
9
1, 0, 0, 1, 3, 16, 96, 675, 5413, 48800, 488592, 5379333, 64595975, 840192288, 11767626752, 176574062535, 2825965531593, 48052401132800, 865108807357216, 16439727718351881, 328839946389605643, 6906458590966507696 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Permanent of the (0,1)-matrix having (i,j)-th entry equal to 0 iff this is on the diagonal or the first upper-diagonal. - Simone Severini, Oct 14 2004

REFERENCES

W. Ahrens, Mathematische Unterhaltungen und Spiele. Teubner, Leipzig, Vol. 1, 3rd ed., 1921; Vol. 2, 2nd ed., 1918. See Vol. 2, p. 79.

J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff.

M. Wyman and L. Moser, On the probleme des menages, Canad. J. Math., 10 (1958), 468-480.

LINKS

T. D. Noe, Table of n, a(n) for n=0..100

W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig: B. G. Teubner, 1901.

FORMULA

a(n) = (n - 1) a(n - 2) + (n - 1) a(n - 1) + a(n - 3).

Contribution from Paul Barry, Feb 08 2009: (Start)

G.f.: 1/(1+x-x/(1+x-x/(1+x-2x/(1+x-2x/(1+x-3x/(1+x-3x/(1+x-4x/(1+..... (continued fraction);

a(n) = sum{k=0..n, C(2n-k,k)*(n-k)!*(-1)^k} (End)

a(n) = (-1)^n*hypergeom([1, -n, n+1],[1/2],1/4). - Mark van Hoeij, Nov 12 2009

a(n) = round( 2*exp(-2)*(BesselK(1+n,2)+BesselK(n,2)) ) for n>0. - Mark van Hoeij, Nov 12 2009

a(n) = sum{k=0..n, (-1)^(n-k)*C(n+k,2k)k!}. - Paul Barry, Jun 23 2010

G.f.: sum_{n>=0} n!*x^n/(1+x)^(2*n+1). - Ira M. Gessel, Jan 15 2013

a(n) ~ exp(-2)*n!. - Vaclav Kotesovec, Mar 10 2014

a(-1 - n) = -a(n). - Michael Somos, May 28 2014

EXAMPLE

G.f. = 1 + x^3 + 3*x^4 + 16*x^5 + 96*x^6 + 675*x^7 + 5413*x^8 + ...

MAPLE

V := proc(n) local k; add( binomial(2*n-k, k)*(n-k)!*(x-1)^k, k=0..n); end; W := proc(r, s) coeff( V(r), x, s ); end; A000271 := n->W(n-2, 0);

MATHEMATICA

f[n_] := Sum[(-1)^(n - k) k! Binomial[n + k, 2 k], {k, 0, n}]; Array[f, 22, 0] (* Jean-François Alcover, Apr 11 2011, after Paul Barry *)

RecurrenceTable[{a[0]==1, a[1]==a[2]==0, a[n]==(n-1)a[n-2]+(n-1)a[n-1]+ a[n-3]}, a, {n, 30}] (* Harvey P. Dale, Jun 01 2012 *)

a[ n_] := (-1)^n HypergeometricPFQ[{1, -n, n + 1}, {1/2}, 1/4]; (* Michael Somos, May 28 2014 *)

PROG

(MAGMA) [ &+[(-1)^(n-k)*Binomial(n+k, 2*k)*Factorial(k): k in [0..n]]: n in [0..21]]; // Bruno Berselli, Apr 11 2011

CROSSREFS

Cf. A000179. A diagonal of A058057.

Sequence in context: A006347 A000270 A157051 * A157016 A228792 A233203

Adjacent sequences:  A000268 A000269 A000270 * A000272 A000273 A000274

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from James A. Sellers, Aug 21 2000

More terms from Simone Severini, Oct 14 2004

STATUS

approved

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Last modified July 26 13:19 EDT 2014. Contains 244946 sequences.