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A001909
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a(n) = n*a(n-1) + (n-4)*a(n-2).
(Formerly M3576 N1450)
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15
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0, 1, 4, 21, 134, 1001, 8544, 81901, 870274, 10146321, 128718044, 1764651461, 25992300894, 409295679481, 6860638482424, 121951698034461, 2291179503374234, 45361686034627361, 943892592746534964
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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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 Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Jaap Spies (j.spies(AT)hccnet.nl), 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(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 ( Febr 27 2010). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010 ]
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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).
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), p. 197-210.
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FORMULA
| E.g.f.: exp(-x)/(1-x)^5 = sum_{n>=0} a(n+3)/n! x^n. - Michael Somos, Feb 19 2003
G.f.: x*hypergeom([1,5],[],x/(x+1))/(x+1) - Mark van Hoeij, Nov 07 2011
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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/(1-x)^4). This adds up as 9 + 4*2*4 + (6*1)*20 + 840 = 1001 = b(4) = A001909(7). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010]
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PROG
| (PARI) a(n)=if(n<2, 0, -contfracpnqn(matrix(2, n, i, j, j-4*(i==1)))[1, 1])
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CROSSREFS
| Cf. A000255, A000153, A000261, A001910, A090010, A055790, A090012-A090016.
a(n) = A086764(n+1,4), n>=2. A000261 (necklaces and three cords). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010]
Sequence in context: A131965 A104982 A195440 * A205077 A052852 A121124
Adjacent sequences: A001906 A001907 A001908 * A001910 A001911 A001912
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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