A000426 Coefficients of ménage hit polynomials.

0, 1, 1, 1, 8, 35, 211, 1459, 11584, 103605, 1030805, 11291237, 135015896, 1749915271, 24435107047, 365696282855, 5839492221440, 99096354764009, 1780930394412009, 33789956266629001, 674939337282352360, 14157377139256183723, 311135096550816014651

Offset: 1

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn
• easy

References

• J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 198.
• 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).
• H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff.

Links

• David W. Wilson, Table of n, a(n) for n = 1..100
• R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976
• H. M. Taylor, A problem on arrangements, Mess. Math., 32 (1902), 60ff. [Annotated scanned copy]

Crossrefs

Cf. A000179, A000271. A diagonal of A058057.

Programs

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

Mathematica

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 *)

Formula

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

a(n) = A000033(n)/n.

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.

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

a(n) ~ 2/e^2*(n-1)!. - Vaclav Kotesovec, Oct 26 2012

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

a(n)+2*a(n+p)+a(n+2*p) is divisible by p for any prime p. - Mark van Hoeij, Jun 13 2019

Extensions

Edited by David W. Wilson, Dec 27 2007