1,1

Row sums are:

{3, -12, 48, -192, 768, -3072, 12288, -49152, 196608, -786432, 3145728}.

P. J. Olver, Classical Invariant Theory, Cambridge Univ. Press, p. 242.

McKean and Moll, Elliptic Curves, Function Theory,Geometry, Arithmetic, Cambridge University Press, New York, 199, page 172

b(x,y,n)=Sum[Binomial[n,i]*x^i*y^(n-i),{i,0,n}]; transforms: x'->(a1*x + b1)/(c1*x + d1); y'->(a2*y + b2)/(c2*y + d2); b1(x,y,n)=(c1*x + b1)^(k)*(c2*y + d2)^(k)*b(x',y',n); f(x,y,z,n)=b1(x,y,n)+b1(y,z,n)+b1(z,x,n); Out_n,m=Coefficients(f(x,y,z,n)).

{3},

{-16,4},

{88, -48, 8},

{-496, 432, -144, 16},

{2848, -3456, 1728, -384, 32},

{-16576, 25920, -17280, 5760, -960,64},

{97408, -186624, 155520, -69120, 17280, -2304, 128},

{-576256, 1306368, -1306368, 725760, -241920, 48384, -5376, 256},

{3424768, -8957952,10450944, -6967296, 2903040, -774144, 129024, -12288, 512}, {-20417536, 60466176, -80621568, 62705664, -31352832, 10450944, -2322432, 331776, -27648, 1024},

{121980928, -403107840, 604661760, -537477120, 313528320, -125411328, 34836480, -6635520, 829440, -61440, 2048}

a1 = 1; b1 = -2; c1 = 0; d1 = 1; a2 = 0; b2 = 1; c2 = 1; d2 = -2; p[x_, y_, k_] = (c1*x + b1)^(k)*(c2*y + d2)^(k)*Sum[Binomial[k, i]*((a1*x + b1)/(c1*x + d1))^i*((a2*y + b2)/(c2*y + d2))^(k - i), {i, 0, k}]; f[x_, y_, z_, k_] = p[x, y, k] + p[y, z, k] + p[z, x, k]; Table[ExpandAll[f[x, y, z, k]], {k, 0, 10}]; a = Table[CoefficientList[f[x, y, z, k] /. y -> 1 /. z -> 1, x], {k, 0, 10}]; Flatten[a] Table[Apply[Plus, CoefficientList[f[x, y, z, k] /. y -> 1 /. z -> 1, x]], {k, 0, 10}]

Sequence in context: A078355 A107823 A378639 * A165969 A098373 A054793

Adjacent sequences: A139812 A139813 A139814 * A139816 A139817 A139818

uned,tabf,sign

Roger L. Bagula, Jun 14 2008

approved