A000426 Coefficients of ménage hit polynomials. (Formerly M4515 N1910)

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

Offset

1,5

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]

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

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 *)

Prog

(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

Crossrefs

Cf. A000179, A000271. A diagonal of A058057.

Sequence in context: A223901 A192257 A297609 * A250889 A200312 A339325

Adjacent sequences: A000423 A000424 A000425 * A000427 A000428 A000429

Keyword

nonn,easy

Author

N. J. A. Sloane and Simon Plouffe

Extensions

Edited by David W. Wilson, Dec 27 2007

Status

approved