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Search: id:A138781
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| A138781 |
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Triangle read by rows: coefficients of polynomials arising in the spontaneous magnetization of the anisotropic square lattice Ising model (see pp. 174-5 of the Guttmann reference). |
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1
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| 1, 2, 3, 2, 3, 16, 32, 16, 3, 4, 46, 200, 305, 200, 46, 4, 5, 100, 770, 2380, 3472, 2380, 770, 100, 5, 6, 185, 2230, 11600, 30240, 41244, 30240, 11600, 2230, 185, 6, 7, 308, 5362, 42140, 172795, 393008, 515332, 393008, 172795, 42140, 5362, 308, 7
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OFFSET
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1,2
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COMMENT
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Row n has 2n-1 terms.
Sum of entries in row n is A097184(n-1).
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REFERENCES
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A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.
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FORMULA
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The row generating polynomial P[n,y] of row n is defined by 1-M(x,y)=2*y*Sum(P[n,y]/(1-y)^(2n)*x^n, n=1..infinity), where M(x,y)=(1-16xy/[(1-x)^2*(1-y)^2])^(1/8).
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EXAMPLE
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Triangle starts:
1;
2,3,2;
3,16,32,16,3;
4,46,200,305,200,46,4
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MAPLE
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M:=(1-16*x*y/((1-x)^2*(1-y)^2))^(1/8): oneminusM:=simplify(series(1-M, x=0, 10)): for n to 7 do P[n]:=sort((1/2)*(y-1)^(2*n)*coeff(oneminusM, x, n)/y) end do: for n to 7 do seq(coeff(P[n], y, k), k=0..2*n-2) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A097184.
Sequence in context: A077942 A077989 A109620 this_sequence A038700 A125767 A131580
Adjacent sequences: A138778 A138779 A138780 this_sequence A138782 A138783 A138784
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 11 2008
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