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%I M4515 N1910 #44 Sep 08 2022 08:44:28
%S 0,1,1,1,8,35,211,1459,11584,103605,1030805,11291237,135015896,
%T 1749915271,24435107047,365696282855,5839492221440,99096354764009,
%U 1780930394412009,33789956266629001,674939337282352360,14157377139256183723,311135096550816014651
%N Coefficients of ménage hit polynomials.
%D J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
%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).
%D H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff.
%H David W. Wilson, <a href="/A000426/b000426.txt">Table of n, a(n) for n = 1..100</a>
%H R. C. Read, <a href="/A000684/a000684_1.pdf">Letter to N. J. A. Sloane, Oct. 29, 1976</a>
%H H. M. Taylor, <a href="/A000179/a000179.pdf">A problem on arrangements</a>, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]
%F a(n) = Sum_{k=2..n} (-1)^k*(2n-k-1)!*(n-k)!/((2n-2k)!*(k-2)!).
%F a(n) = A000033(n)/n.
%F a(n) = ((2*n-5)*a(n-1) + (5*n-11)*a(n-2) + (5*n-14)*a(n-3) + (2*n-5)*a(n-4) + 2*a(n-5))/2 for n >= 6.
%F Shorter recurrence: (14*n-67)*a(n) = (14*n^2-95*n+137)*a(n-1) + (14*n^2-105*n+180)*a(n-2) - 24*a(n-4) + (57-10*n)*a(n-3). - _Vaclav Kotesovec_, Oct 26 2012
%F a(n) ~ 2/e^2*(n-1)!. - _Vaclav Kotesovec_, Oct 26 2012
%F a(n) = round((exp(-2)*(8*BesselK(n,2) - (4*n-10)*BesselK(n-1,2)))) for n > 6. - _Mark van Hoeij_, Jun 09 2019
%F a(n)+2*a(n+p)+a(n+2*p) is divisible by p for any prime p. - _Mark van Hoeij_, Jun 13 2019
%t Table[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 (Magma) [0] cat [&+[(-1)^k*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 Cf. A000179, A000271. A diagonal of A058057.
%K nonn,easy
%O 1,5
%A _N. J. A. Sloane_ and _Simon Plouffe_
%E Edited by _David W. Wilson_, Dec 27 2007
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