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0, 0, 1, 3, 12, 50, 225, 1092, 5684, 31572, 186300, 1163085, 7654350, 52928460, 383437327, 2902665885, 22907918640, 188082362120, 1603461748491, 14169892736484, 129594593170210, 1224875863061970, 11948280552370932
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Number of blocks of size 2 in all set partitions of {1,2,...,n}. Example: a(3)=3 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, containing exactly 3 blocks of size 2. a(n)=Sum(k*A124498(n-1,k), k>=0}. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 06 2006
Number of partitions of {1...n} containing 2 pairs of consecutive integers, where each pair is counted within a block and a string of more than 2 consecutive integers are counted two at a time. E.g. a(4) = 3 because the partitions of {1,2,3,4} with 2 pairs of consecutive integers are 123/4,12/34,1/234. - A. O. Munagi (amunagi(AT)yahoo.com), Apr 10 2005
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REFERENCES
| A. O. Munagi, Set Partitions with Successions and Separations, Int. J. Math and Math. Sc. 2005, no. 3 (2005), 451-463.
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LINKS
| A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451-463.
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FORMULA
| a(n) = binomial(n-1, 2)*Bell(n-3), the case r = 2 of the general case of r pairs: c(n, r) = binomial(n-1, r)B(n-r-1).
E.g.f.: z^2/2 * e^(e^z-1) - Frank Ruskey (ruskey(AT)cs.uvic.ca), Dec 26 2006
G.f.: exp(-1)*Sum(x^2/(n!*(1-n*x)^3),n=0..infinity). - Vladeta Jovovic (vladeta(AT)eunet.rs), Feb 05 2008
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=2, a(n)=(-1)^(n-2)coeff(charpoly(A,x),x^2). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
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MAPLE
| [seq(binomial(n, 2)*combinat[bell](n-2), n=0..50)];
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CROSSREFS
| Cf. A105480, A105489, A105484, A124498.
Sequence in context: A151178 A151179 A191242 * A151180 A151181 A094601
Adjacent sequences: A105476 A105477 A105478 * A105480 A105481 A105482
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KEYWORD
| easy,nonn
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AUTHOR
| A. O. Munagi (amunagi(AT)yahoo.com), Apr 10 2005
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 01 2007
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