A138781 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).

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

Offset

1,2

Comments

Row n has 2n-1 terms.

Sum of entries in row n is A097184(n-1).

References

A. J. Guttmann, Indicators of solvability for lattice models, Discrete Math., 217 (2000), 167-189.

Links

Table of n, a(n) for n=1..49.

Formula

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).

Example

Triangle starts:
1;
2,3,2;
3,16,32,16,3;
4,46,200,305,200,46,4

Maple

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

Crossrefs

Cf. A097184.

Sequence in context: A077942 A077989 A109620 * A224416 A282049 A038700

Adjacent sequences: A138778 A138779 A138780 * A138782 A138783 A138784

Keyword

nonn,tabf

Author

Emeric Deutsch, May 11 2008

Status

approved