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Search: id:A100862
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| A100862 |
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Triangle read by rows: T(n,k) is the number of k-matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'. |
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+0
6
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| 1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 21, 10, 1, 1, 15, 55, 55, 15, 1, 1, 21, 120, 215, 120, 21, 1, 1, 28, 231, 665, 665, 231, 28, 1, 1, 36, 406, 1736, 2835, 1736, 406, 36, 1, 1, 45, 666, 3990, 9891, 9891, 3990, 666, 45, 1, 1, 55, 1035, 8310, 29505, 45297, 29505, 8310, 1035, 55, 1, 1, 66, 1540, 16005, 77715, 173712, 173712, 77715, 16005, 1540, 66, 1
(list; table; graph; listen)
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OFFSET
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0,5
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COMMENT
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Row n has n+1 terms. Row sums yield A005425. T(n,k)=T(n,n-k).
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FORMULA
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E.g.f.=exp(z+tz+tz^2/2). Row generating polynomial=P[n]=[ -i*sqrt(t/2)]^n*H(n, i(1+t)/sqrt(2t)), where H(n, x) is a Hermite polynomial and i=sqrt(-1). Row generating polynomials P[n] satisfy P[0]=1, P[n]=(1+t)P[n-1]+(n-1)tP[n-2].
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EXAMPLE
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T(3,2)=6 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following six 2-matchings: {Aa,BC},{Bb,AC},{Cc,AB},{Aa,Bb},{Aa,Cc} and {Bb,Cc}.
Triangle starts:
1;
1,1;
1,3,1;
1,6,6,1;
1,10,21,10,1;
1,15,55,55,15,1;
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MAPLE
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P[0]:=1: for n from 1 to 11 do P[n]:=sort(expand((1+t)*P[n-1]+(n-1)*t*P[n-2])) od: for n from 0 to 11 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # gives the sequence in triangular form
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PROGRAM
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(PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); n!*polcoeff(polcoeff(exp(X+Y*X^2/2+X*Y), n, x), k, y)} (Paul Hanna)
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CROSSREFS
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Cf. A005425.
Cf. A107102 (matrix inverse).
Sequence in context: A162747 A107105 A088925 this_sequence A098568 A131235 A157243
Adjacent sequences: A100859 A100860 A100861 this_sequence A100863 A100864 A100865
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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), Jan 08 2005
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