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Search: id:A097898
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| A097898 |
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Triangle read by rows: T(n,k) is the number of permutations of [n] with k runs of length 1. For example, 457/3/26/1 has two runs of length 1: 3 and 1. |
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+0
1
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| 1, 0, 1, 1, 0, 1, 1, 4, 0, 1, 6, 6, 11, 0, 1, 19, 51, 23, 26, 0, 1, 109, 212, 269, 72, 57, 0, 1, 588, 1571, 1419, 1140, 201, 120, 0, 1, 4033, 10470, 13343, 7432, 4272, 522, 247, 0, 1, 29485, 87672, 107853, 87552, 33683, 14841, 1291, 502, 0, 1, 246042, 763612
(list; table; graph; listen)
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OFFSET
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0,8
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REFERENCES
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Ira. M. Gessel, Generating functions and enumeration of sequences, Ph. D. Thesis, MIT, 1977.
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FORMULA
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E.g.f.: Bexp(-Ax)/[A*sinh(Bx)+B*cosh(Bx)-sinh(Bx)], where A=(1-t)/2 and B=(1/2)sqrt(t^2+2t-3).
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EXAMPLE
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Triangle starts:
1;
0,1;
1,0,1;
1,4,0,1;
6,6,11,0,1;
19,51,23,26,0,1
Row n has n+1 terms.
T(3,0)=1, T(3,1)=4, T(3,2)=0 and T(3,3)=1 because we have 123, 13(2), (2)13, 23(1), (3)12, (3)(2)(1), the runs of length 1 being shown between parantheses.
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MAPLE
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A:=(1-t)/2: B:=sqrt(t^2+2*t-3)/2: G:=B/exp(A*z)/(A*sinh(B*z)+B*cosh(B*z)-sinh(B*z)): Gserz:=simplify(series(G, z=0, 12)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(n!*coeff(Gserz, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..n+1), n=0..10);
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CROSSREFS
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Sequence in context: A021253 A136586 A092746 this_sequence A154884 A059678 A079642
Adjacent sequences: A097895 A097896 A097897 this_sequence A097899 A097900 A097901
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu) and Ira Gessel (gessel(AT)brandeis.edu), Sep 03 2004
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