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A000159 Coefficients of ménage hit polynomials.

%I A000159 M1834 N0728 #50 Jun 10 2019 23:15:14
%S A000159 2,8,20,152,994,7888,70152,695760,7603266,90758872,1174753372,
%T A000159 16386899368,245046377410,3910358788256,66323124297872,
%U A000159 1191406991067168,22596344660865282,451208920617687720,9461897733571886372,207894669895136763704,4776019866458134139042
%N A000159 Coefficients of ménage hit polynomials.
%D A000159 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.
%D A000159 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000159 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000159 Sean A. Irvine, <a href="/A000159/b000159.txt">Table of n, a(n) for n = 3..250</a>
%H A000159 Belgacem Bouras, <a href="http://www.emis.de/journals/JIS/VOL16/Bouras/bouras4.html">A New Characterization of Catalan Numbers Related to Hankel Transforms and Fibonacci Numbers</a>, Journal of Integer Sequences, 16 (2013), #13.3.3.
%H A000159 M. Dougherty, C. French, B. Saderholm, W. Qian, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/French/french2.html">Hankel Transforms of Linear Combinations of Catalan Numbers</a>, J. Int. Seq. 14 (2011) # 11.5.1.
%F A000159 Conjecture: 2*(-252307*n + 1041077)*a(n) + (504614*n^2 - 3362985*n + 5118150)*a(n-1) + (1280831*n^2 - 7397886*n + 6461565)*a(n-2) + (746598*n^2 - 2913543*n - 1336090)*a(n-3) + (-405481*n^2 + 6175011*n - 15469320)*a(n-4) + (-375862*n^2 + 4098537*n - 8846430)*a(n-5) + 2*(-187931*n + 560630)*a(n-6) = 0. - _R. J. Mathar_, Nov 02 2015
%F A000159 a(n) = round(2*n*(4*exp(-2)*((n+3/2)*BesselK(n-1,2) - (n-9/2)*BesselK(n-2,2)) + (-1)^n)/3) for n > 11 assuming the recurrence is correct. - _Mark van Hoeij_, Jun 09 2019
%F A000159 Conjecture: a(n) + 2*a(n+p) + a(n+2*p) is divisible by p for any prime p except 3. - _Mark van Hoeij_, Jun 10 2019
%Y A000159 A diagonal of A058087.
%Y A000159 Cf. A000179, A000425.
%K A000159 nonn
%O A000159 3,1
%A A000159 _N. J. A. Sloane_, _Simon Plouffe_