A000159 Coefficients of ménage hit polynomials.

2, 8, 20, 152, 994, 7888, 70152, 695760, 7603266, 90758872, 1174753372, 16386899368, 245046377410, 3910358788256, 66323124297872, 1191406991067168, 22596344660865282, 451208920617687720, 9461897733571886372, 207894669895136763704, 4776019866458134139042

Offset: 3

Author

N. J. A. Sloane, Simon Plouffe

Keywords

• nonn

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 197.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Sean A. Irvine, Table of n, a(n) for n = 3..250
• Belgacem Bouras, A New Characterization of Catalan Numbers Related to Hankel Transforms and Fibonacci Numbers, Journal of Integer Sequences, 16 (2013), #13.3.3.
• M. Dougherty, C. French, B. Saderholm, W. Qian, Hankel Transforms of Linear Combinations of Catalan Numbers, J. Int. Seq. 14 (2011) # 11.5.1.

Crossrefs

A diagonal of A058087.

Cf. A000179, A000425.

Formula

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

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

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