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

0, 2, 3, 4, 40, 210, 1477, 11672, 104256, 1036050, 11338855, 135494844, 1755206648, 24498813794, 366526605705, 5851140525680, 99271367764480, 1783734385752162, 33837677493828171, 675799125332580020, 14173726082929399560, 311462297063636041906

Offset

1,2

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

David W. Wilson, Table of n, a(n) for n = 1..100

Formula

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).

a(n) = Sum_{k=2..n} (-1)^k*n*(2n-k-1)!*(n-k)!/((2n-2k)!*(k-2)!). - David W. Wilson, Jun 22 2006

a(n) = n*A000426(n) - Vladeta Jovovic, Dec 27 2007

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

a(n) ~ 2/e^2*n!. - Vaclav Kotesovec, Oct 26 2012

From Mark van Hoeij, Jun 09 2019: (Start)

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

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

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

Mathematica

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 *)

Prog

(Haskell)
fac = a000142
a n = sum $ map f [2..n]
  where f k = g k `div` h k
        g k = (-1)^k * n * fac (2*n-k-1) * fac (n-k)
        h k = fac (2*n-2*k) * fac (k-2)
-- James Spahlinger, Oct 08 2012
(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

Crossrefs

A diagonal of A058087.

Cf. A000179, A000425.

Sequence in context: A037326 A024634 A134305 * A270350 A060411 A037322

Adjacent sequences: A000030 A000031 A000032 * A000034 A000035 A000036

Keyword

nonn,easy

Author

N. J. A. Sloane and Simon Plouffe

Extensions

Extended to 34 terms by N. J. A. Sloane, May 25 2005

Edited and further extended by David W. Wilson, Dec 27 2007

Status

approved