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A052618
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E.g.f. 1/((1-x)^2*(1-x^2)).
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5
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1, 2, 8, 36, 216, 1440, 11520, 100800, 1008000, 10886400, 130636800, 1676505600, 23471078400, 348713164800, 5579410636800, 94152554496000, 1694745980928000, 32011868528640000, 640237370572800000, 13380961044971520000, 294381142989373440000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Permanent of the n X n (0, 1)-matrix with (i, j)-th entry equal to 0 iff (i=1, j=n), (i=2, j=1), (i=3, j=n), (i=4, j=1), ... - Simone Severini, Oct 17 2004
Contribution from Emeric Deutsch, Dec 14 2008: (Start)
a(n) is the number of runs of odd entries in all permutations of {1,2,...,n+1}. Example: a(2)=8 because in the permutations 123,132,213,231,312 and 321 we have a total of 2+1+1+1+1+2 runs of odd entries.
a(n) = Sum(k*A152666(n+1,k),k>=1). (End)
a(n) is the number of permutations of [n+2] whose first place is even and last place is odd (or any equivalent definition with two separate places in a permutation). [From Olivier Gerard, Nov 7 2011]
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LINKS
| INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 563
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FORMULA
| E.g.f.: -1/(-1+x)^2/(-1+x^2).
Recurrence: {a(0)=1, a(1)=2, (-n^2-5*n-4)*a(n)+a(n+2)-2*a(n+1)=0.}.
a(n) = (1/8*(-1)^(-n)+1/4*n^2+n+7/8)*n! for n>0.
a(n) = n!*floor((n+2)/2)*ceil((n+2)/2). [From Emeric Deutsch, Dec 14 2008]
a(n) = n!*A002620(n+2). - R. J. Mathar, Nov 27 2011
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EXAMPLE
| The a(2) = 8 permutations of [4] starting by an even number and ending by an odd number are: 2143, 2341, 2413, 2431, 4123, 4213, 4231, 4321.
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MAPLE
| spec := [S, {S=Prod(Sequence(Z), Sequence(Z), Sequence(Prod(Z, Z)))}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
a := proc (n) options operator, arrow: factorial(n)*floor((1/2)*n+1)*ceil((1/2)*n+1) end proc; seq(a(n), n = 0 .. 20); # Emeric Deutsch, Dec 14 2008
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CROSSREFS
| Cf. A152666. [From Emeric Deutsch, Dec 14 2008]
Sequence in context: A129044 A052582 A020021 * A055142 A183908 A046814
Adjacent sequences: A052615 A052616 A052617 * A052619 A052620 A052621
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KEYWORD
| easy,nonn
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AUTHOR
| encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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