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Search: id:A124302
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| A124302 |
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Number of set partitions of length <=3; sum of first 3 columns of triangle of Stirling numbers of 2nd kind; dimension of space of symmetric polynomials in 3 noncommuting variables. |
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+0
5
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| 1, 1, 2, 5, 14, 41, 122, 365, 1094, 3281, 9842, 29525, 88574, 265721, 797162, 2391485, 7174454, 21523361, 64570082, 193710245, 581130734, 1743392201, 5230176602, 15690529805, 47071589414, 141214768241, 423644304722, 1270932914165
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Positive version of A123183
Row sums of triangle in A056241 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2006
Row sums of triangle in A147746 . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2008]
Hankel transform is := [1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 04 2008]
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REFERENCES
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N. Bergeron, C. Reutenauer, M. Rosas, M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, to appear Canad. J. Math., arXiv:math.CO/0502082
M. Rosas and B. Sagan, Symmetric Functions in Noncommuting Variables. Transactions of the American Mathematical Society, 358 (2006), no. 1, 215-232.
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FORMULA
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O.g.f. (q^2 - 3*q + 1)/(3*q^2 - 4*q + 1) = sum(q^k/prod((1-i*q),i=1..k),k=0..3) a(n) = 4*a(n-1)-3*a(n-2); a(0) = 1, a(1) = 1, a(2) = 2 a(n) = add(A008277(n,k),k=1..3)
Inverse binomial transform of A007581 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2006
a(n)=Sum_{k, 0<=k<=n}A056241(n,k), n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2006
a(0)=1, a(n)=(3^(n-1)+1)/2 for n>=1, see A007051 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 30 2006
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EXAMPLE
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There are 15 set partitions of {1,2,3,4}, only {{1},{2},{3},{4}} has length >3 so a(4) = 14
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MAPLE
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a:=proc(n); if n<3 then [1, 1, 2][n+1]; else 4*a(n-1)-3*a(n-2); fi; end:
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CROSSREFS
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Cf. A123183, A001519, A000110, A008277.
Sequence in context: A116845 A116849 A007051 this_sequence A123183 A088355 A113485
Adjacent sequences: A124299 A124300 A124301 this_sequence A124303 A124304 A124305
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KEYWORD
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nonn
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AUTHOR
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Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 25 2006
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