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A000271
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Sums of menage numbers.
(Formerly M3020 N1222)
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9
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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
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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. 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.
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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.
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FORMULA
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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). [From Mark van Hoeij, Nov 12 2009]
a(n) = round( 2*exp(-2)*(BesselK(1+n,2)+BesselK(n,2)) ) for n>0. [From Mark van Hoeij, Nov 12 2009]
a(n) = sum{k=0..n, (-1)^(n-k)*C(n+k,2k)k!}. [From Paul Barry, Jun 23 2010]
G.f.: sum_{n>=0} n!*x^n/(1+x)^(2*n+1). [From Ira M. Gessel, Jan 15 2013]
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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] (* From 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 *)
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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 A074553 A193037
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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