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

Author

Emeric Deutsch, May 11 2008

Keywords

• nonn
• tabf

Comments

Row n has 2n-1 terms.

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

Examples

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

References

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

Crossrefs

Cf. A097184.

Programs

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

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).