A000033 Coefficients of ménage hit polynomials.

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

Offset: 1

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn
• easy

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

Crossrefs

A diagonal of A058087.

Cf. A000159, A000179, A000425.

Programs

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

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 *)

SageMath

def A000033(n): return n*sum((-1)^k*(2*n-3-2*k)*factorial(n-k-2)*binomial(2*n-k-3, k) for k in range(n-1)) # G. C. Greubel, Jul 10 2025

Formula

a(n) = coefficient of t^2 in polynomial p(t) = Sum_{k=0..n} 2*n*C(2*n-k,k)*(n-k)!*(t-1)^k/(2*n-k).

a(n) = Sum_{k=2..n} (-1)^k*n*(2*n-k-1)!*(n-k)!/((2*n-2*k)!*(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

Extensions

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

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