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

%I M0602 N0216 #52 Sep 08 2022 08:44:27

%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} 2*n*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

%F From _Mark van Hoeij_, Jun 09 2019: (Start)

%F a(n) = round(2*(exp(-2)*n*(4*BesselK(n,2)-(2*n-5)*BesselK(n-1,2))-(-1)^n)) for n > 9.

%F a(n) = (3/2)*(A000159(n+1)*n/(n+1) - A000159(n))/(n-1) for n > 2. (End)

%F Conjecture: a(n) + 2*a(n+p) + a(n+2*p) is divisible by p for any prime p. - _Mark van Hoeij_, Jun 10 2019

%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

%o (Magma) [0] cat [&+[(-1)^k*n*Factorial(2*n-k-1)*Factorial(n-k)/(Factorial(2*n-2*k)*Factorial(k-2)): k in [2..n]]: n in [2..25]]; // _Vincenzo Librandi_, Jun 11 2019

%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 March 28 07:46 EDT 2024. Contains 371235 sequences. (Running on oeis4.)