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A000033 Coefficients of ménage hit polynomials.
(Formerly M0602 N0216)
6

%I M0602 N0216

%S 0,2,3,4,40,210,1477,11672,104256,1036050,11338855,135494844,

%T 1755206648,24498813794,366526605705,5851140525680,99271367764480,

%U 1783734385752162,33837677493828171,675799125332580020,14173726082929399560,311462297063636041906

%N Coefficients of ménage hit polynomials.

%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.

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

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

%H David W. Wilson, <a href="/A000033/b000033.txt">Table of n, a(n) for n = 1..100</a>

%F a(n) = coefficient of t^2 in polynomial p(t) = SUM(k = 0..n, (2n C(2n-k,k) (n-k)! (t-1)^k)/(2n-k))

%F a(n) = SUM(k = 2..n, ((-1)^k n (2n-k-1)! (n-k)!)/((2n-2k)! (k-2)!)) - _David W. Wilson_, Jun 22 2006

%F a(n) = n*A000426(n) - _Vladeta Jovovic_, Dec 27 2007

%F Recurrence: (n-3)*(n-2)*(2*n-5)*(2*n-7)*a(n) = (n-3)*(n-2)*n*(2*n-7)^2*a(n-1) + (n-4)*(n-3)*n*(2*n-3)^2*a(n-2) + (n-2)*n*(2*n-5)*(2*n-3)*a(n-3). - _Vaclav Kotesovec_, Oct 26 2012

%F a(n) ~ 2/e^2*n!. - _Vaclav Kotesovec_, Oct 26 2012

%t Table[n*Sum[(-1)^k*(2*n-k-1)!*(n-k)!/((2*n-2*k)!*(k-2)!),{k,2,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 26 2012 *)

%o (Haskell)

%o fac = a000142

%o a n = sum $ map f [2..n]

%o where f k = g k `div` h k

%o g k = (-1)^k * n * fac (2*n-k-1) * fac (n-k)

%o h k = fac (2*n-2*k) * fac (k-2)

%o -- _James Spahlinger_, Oct 08 2012

%Y A diagonal of A058087.

%Y Cf. A000179, A000425.

%K nonn,easy

%O 1,2

%A _N. J. A. Sloane_ and _Simon Plouffe_

%E Extended to 34 terms by _N. J. A. Sloane_, May 25 2005

%E Edited and further extended by _David W. Wilson_, Dec 27 2007

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Last modified February 22 07:43 EST 2018. Contains 299447 sequences. (Running on oeis4.)