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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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11
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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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]
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
We observe that the a(n) equal b(2*n) with the b(n) the number of paths of length n starting and ending at the initial node on the path graph P_5, see the second Maple program.
(End)
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REFERENCES
| 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
| 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
E.g.f.:(2+3*exp(x)+exp(3x))/6
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EXAMPLE
| 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
| a:=proc(n); if n<3 then [1, 1, 2][n+1]; else 4*a(n-1)-3*a(n-2); fi; end:
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), May 29 2010: (Start)
with(GraphTheory): G:=PathGraph(5): A:= AdjacencyMatrix(G): nmax:=27; for n from 0 to 2*nmax do B(n):=A^n; b(n):=B(n)[1, 1]; od: for n from 0 to nmax do a(n):=b(2*n) od: seq(a(n), n=0..nmax);
(End)
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MATHEMATICA
| a=Exp[x]-1; Range[0, 20]! CoefficientList[Series[1+a+a^2/2+a^3/6, {x, 0, 20}], x]
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CROSSREFS
| Cf. A123183, A001519, A000110, A008277.
Cf. A038754, A048328 and A068911.
Sequence in context: A116845 A116849 A007051 * A123183 A088355 A113485
Adjacent sequences: A124299 A124300 A124301 * A124303 A124304 A124305
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KEYWORD
| nonn
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AUTHOR
| Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Oct 25 2006
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