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A000271
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Sums of ménage numbers.
(Formerly M3020 N1222)
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11
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1, 0, 0, 1, 3, 16, 96, 675, 5413, 48800, 488592, 5379333, 64595975, 840192288, 11767626752, 176574062535, 2825965531593, 48052401132800, 865108807357216, 16439727718351881, 328839946389605643, 6906458590966507696
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OFFSET
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0,5
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COMMENTS
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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
From Vladimir Shevelev, Jun 21 2015: (Start)
Let 2*n!*V(n) be the number of ways of seating n married couples at 2*n chairs arranged side-by-side in a straight line, men and women in alternate positions, so that no husband is next to his wife.
It is known [Riordan, Ch. 8, Th. 1, t=0] that, if 2*n!*U(n) is a solution of an analogous problem at a circular table, then U(n) = V(n) - V(n-1), n>=3, where U(n) = A000179(n). Thus V(n) = Sum_{i=3,...,n} A000179(i), n>=1, and comparing the initial conditions, we conclude that a(n) = V(n), n>=1. This gives a combinatorial interpretation for 2*n!*a(n).
(End)
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REFERENCES
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W. Ahrens, Mathematische Unterhaltungen und Spiele. Teubner, Leipzig, Vol. 1, 3rd ed., 1921; Vol. 2, 2nd ed., 1918. See Vol. 2, p. 79.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
W. Ahrens, Mathematische Unterhaltungen und Spiele, Leipzig: B. G. Teubner, 1901.
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.
V. Kotesovec, Non-attacking chess pieces, 6ed, 2013, p. 222.
H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]
M. Wyman and L. Moser, On the problème des ménages, Canad. J. Math., 10 (1958), 468-480.
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FORMULA
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a(n) = (n - 1) a(n - 2) + (n - 1) a(n - 1) + a(n - 3).
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
a(n) = Sum_{i=3..n} A000179(i), n>=1. - Vladimir Shevelev, Jun 21 2015
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EXAMPLE
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G.f. = 1 + x^3 + 3*x^4 + 16*x^5 + 96*x^6 + 675*x^7 + 5413*x^8 + ...
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MAPLE
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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);
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MATHEMATICA
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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 *)
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PROG
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(MAGMA) [ &+[(-1)^(n-k)*Binomial(n+k, 2*k)*Factorial(k): k in [0..n]]: n in [0..21]]; // Bruno Berselli, Apr 11 2011
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CROSSREFS
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Cf. A000179. A diagonal of A058057.
Sequence in context: A006347 A000270 A157051 * A157016 A228792 A233203
Adjacent sequences: A000268 A000269 A000270 * A000272 A000273 A000274
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers, Aug 21 2000
More terms from Simone Severini, Oct 14 2004
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STATUS
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approved
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