OFFSET
3,3
COMMENTS
With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=5 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, pp. 201-202. - Jaap Spies, Dec 12 2003
a(n+4)=:b(n), n>=1, enumerates the ways to distribute n beads labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and k=5 indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as a beadless cords contribute each a factor 1 in the counting, e.g., b(0):= 1*1 =1. See A000255 for the description of a fixed cord with beads.
This produces for b(n) the exponential (aka binomial) convolution of the subfactorial sequence {A000166(n)} and the sequence {A001720(n+4) = (n+4)!/4!}. See the necklaces and cords problem comment in A000153. Therefore also the recurrence b(n) = (n+4)*b(n-1) + (n-1)*b(n-2) with b(-1)=0 and b(0)=1 holds. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 02 2010
REFERENCES
Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 188.
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
T. D. Noe, Table of n, a(n) for n = 3..100
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), pp. 197-210.
FORMULA
a(n) = A086764(n+1,5), n>=3.
E.g.f. with offset -1: (exp(-x)/(1-x))*(1-x)^5 = exp(-x)/(1-x)^6. - Wolfdieter Lang, Jun 02 2010
G.f.: x*hypergeom([1,6],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
a(n) = hypergeometric([6,-n+4],[],1)*(-1)^n for n >=4. - Peter Luschny, Sep 20 2014
EXAMPLE
Necklaces and 5 cords problem. For n=4 one considers the following weak 2 part compositions of 4: (4,0), (3,1), (2,2), and (0,4), where (1,3) does not appear because there are no necklaces with 1 bead. These compositions contribute respectively sf(4)*1, binomial(4,3)*sf(3)*c5(1), (binomial(4,2)*sf(2))*c5(2), and 1*c5(4) with the subfactorials sf(n):=A000166(n) (see the necklace comment there) and the c5(n):=A001720(n+4) numbers for the pure 5 cord problem (see the remark on the e.g.f. for the k cords problem in A000153; here for k=5: 1/(1-x)^5). This adds up as 9 + 4*2*5 + (6*1)*30 + 1680 = 1909 = b(4) = A001910(8). - Wolfdieter Lang, Jun 02 2010
MAPLE
a := n -> `if`(n=3, 0, hypergeom([6, -n+4], [], 1))*(-1)^n;
seq(round(evalf(a(n), 100)), n=3..20); # Peter Luschny, Sep 20 2014
MATHEMATICA
t = {0, 1}; Do[AppendTo[t, n*t[[-1]] + (n - 5) t[[-2]]], {n, 5, 20}]; t (* T. D. Noe, Aug 17 2012 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved