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A105483
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Number of partitions of {1...n} containing one string of 3 consecutive integers, counted within a block.
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2
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1, 2, 8, 32, 141, 672, 3451, 18962, 110882, 686866, 4489422, 30853656, 222276063, 1674067342, 13149209956, 107481488424, 912490408782, 8031867965568, 73181346933680, 689194657064660, 6699707386510583, 67143409071264516
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OFFSET
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3,2
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REFERENCES
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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
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Table of n, a(n) for n=3..24.
A. O. Munagi, Set Partitions with Successions and Separations,IJMMS 2005:3 (2005),451-463.
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FORMULA
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a(n)=Sum(c(n, k, 1), k=1...n), where c(n, k, 1) is the case r =1 of c(n, k, r) given by c(n, k, r)=c(n-1, k-1, r)+(k-1)c(n-1, k, r)+c(n-2, k-1, r)+(k-1)c(n-2, k, r)+c(n-1, k, r-1)-c(n-2, k-1, r-1)-(k-1)c(n-2, k, r-1), r=0, 1, .., n-k-1, k=1, 2, .., n-2r, c(n, k, 0)=sum(binomial(n-j, j)*S2(n-j-1, k-1), j= 0..floor(n/2)).
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EXAMPLE
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a(5) = 8 because the partitions of {1,2,3,4,5} with one 3-string of consecutive integers are 1235/4, 1345/2, 15/234, 123/45, 12/345, 123/4/5, 1/234/5, 1/2/345.
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MAPLE
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c := proc(n, k, r) option remember ; local j ; if r =0 then add(binomial(n-j, j)*combinat[stirling2](n-j-1, k-1), j=0..floor(n/2)) ; else if r <0 or r > n-k-1 then RETURN(0) fi ; if n <1 then RETURN(0) fi ; if k <1 then RETURN(0) fi ; RETURN( c(n-1, k-1, r)+(k-1)*c(n-1, k, r)+c(n-2, k-1, r)+(k-1)*c(n-2, k, r) +c(n-1, k, r-1)-c(n-2, k-1, r-1)-(k-1)*c(n-2, k, r-1) ) ; fi ; end: A105483 := proc(n) local k ; add(c(n, k, 1), k=1..n) ; end: for n from 3 to 26 do printf("%d, ", A105483(n)) ; od ; # R. J. Mathar, Feb 20 2007
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CROSSREFS
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Cf. A105484, A105489, A105493.
Sequence in context: A150854 A150855 A150856 * A150857 A150858 A150859
Adjacent sequences: A105480 A105481 A105482 * A105484 A105485 A105486
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KEYWORD
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nonn
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AUTHOR
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Augustine O. Munagi, Apr 10 2005
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EXTENSIONS
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More terms from R. J. Mathar, Feb 20 2007
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STATUS
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approved
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