

A001909


a(n) = n*a(n1) + (n4)*a(n2), a(2) = 0, a(3) = 1.
(Formerly M3576 N1450)


19



0, 1, 4, 21, 134, 1001, 8544, 81901, 870274, 10146321, 128718044, 1764651461, 25992300894, 409295679481, 6860638482424, 121951698034461, 2291179503374234, 45361686034627361, 943892592746534964, 20592893110265899381, 470033715095287415734
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OFFSET

2,3


COMMENTS

With offset 1, permanent of (0,1)matrix of size n X (n+d) with d=4 and n zeros not on a line. This is a special case of Theorem 2.3 of SeokZun Song et al. Extremes of permanents of (0,1)matrices, pp. 201202.  Jaap Spies, Dec 12 2003
a(n+3)=: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 four 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 {A001715 (n+3)}. See the necklaces and cords problem comment in A000153. Therefore also the recurrence b(n) = (n+3)*b(n1) + (n1)*b(n2) 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 = 2..100
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7.  From N. J. A. Sloane, Feb 06 2013
SeokZun 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. 197210.


FORMULA

a(n) = A086764(n+1,4), n>=2.
E.g.f.: exp(x) / (1  x)^5 = Sum_{k>=0} a(k+3) * x^k / k!.  Michael Somos, Feb 19 2003
G.f.: x*hypergeom([1,5],[],x/(x+1))/(x+1).  Mark van Hoeij, Nov 07 2011
a(n) = hypergeom([5,n+3],[],1))*(1)^(n+1) for n>=3.  Peter Luschny, Sep 20 2014


EXAMPLE

Necklaces and four 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)*c4(1), (binomial(4,2)*sf(2))*c4(2), and 1*c4(4) with the subfactorials sf(n):=A000166(n) (see the necklace comment there) and the c4(n):=A001715(n+3) = (n+3)!/3! numbers for the pure 4 cord problem (see the remark on the e.g.f. for the k cords problem in A000153; here for k=4: 1/(1x)^4). This adds up as 9 + 4*2*4 + (6*1)*20 + 840 = 1001 = b(4) = A001909(7).  Wolfdieter Lang, Jun 02 2010
x^3 + 4*x^4 + 21*x^5 + 134*x^6 + 1001*x^7 + 8544*x^8 + 81901*x^9 + 870274*x^10 + ...


MAPLE

a := n > `if`(n<4, n2, hypergeom([5, n+3], [], 1))*(1)^(n+1);
seq(round(evalf(a(n), 100)), n=2..22); # Peter Luschny, Sep 20 2014


MATHEMATICA

t = {0, 1}; Do[AppendTo[t, n*t[[1]] + (n4)*t[[2]]], {n, 4, 20}]; t (* T. D. Noe, Aug 17 2012 *)
nxt[{n_, a_, b_}]:={n+1, b, b(n+1)+a(n3)}; NestList[nxt, {3, 0, 1}, 20][[All, 2]] (* Harvey P. Dale, Jul 17 2018 *)


PROG

(PARI) {a(n) = if( n<2, 0, contfracpnqn( matrix(2, n, i, j, j  4*(i==1))) [1, 1])} /* Michael Somos, Feb 19 2003 */


CROSSREFS

Cf. A000255, A000153, A000261, A001910, A090010, A055790, A090012A090016, A086764. A000261 (necklaces and three cords).
Sequence in context: A226067 A104982 A195440 * A205077 A292928 A209881
Adjacent sequences: A001906 A001907 A001908 * A001910 A001911 A001912


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


STATUS

approved



