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A001910
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a(n) = n*a(n-1) + (n-5)*a(n-2).
(Formerly M3965 N1637)
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16
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0, 1, 5, 31, 227, 1909, 18089, 190435, 2203319, 27772873, 378673901, 5551390471, 87057596075, 1453986832381, 25762467303377, 482626240281739, 9530573107600319, 197850855756232465, 4307357140602486869
(list; graph; refs; listen; history; internal format)
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OFFSET
| 3,3
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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, p. 201-202. - Jaap Spies (j.spies(AT)hccnet.nl), 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 ( 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. with offset -1: (exp(-x)/(1-x))*(1-x)^5 = exp(-x)/(1-x)^6. [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010]
G.f.: x*hypergeom([1,6],[],x/(x+1))/(x+1) - Mark van Hoeij, Nov 07 2011
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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). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010]
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CROSSREFS
| Cf. A000255, A000153, A000261, A001909, A001910, A055790, A090012-A090016.
a(n) = A086764(n+1,5), n>=3. A001909 (necklaces and four cords). [From Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Jun 02 2010]
Sequence in context: A199877 A058309 A192950 * A052773 A062147 A069321
Adjacent sequences: A001907 A001908 A001909 * A001911 A001912 A001913
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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