A139815 A triangle of coefficients of a Moebius-transformed Pascal triangle as a sum: 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).

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

Offset

1,1

Comments

Row sums are:

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

References

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

Links

Table of n, a(n) for n=1..43.

Formula

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)).

Example

{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}

Mathematica

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}]

Crossrefs

Sequence in context: A078355 A107823 A378639 * A165969 A098373 A054793

Adjacent sequences: A139812 A139813 A139814 * A139816 A139817 A139818

Keyword

uned,tabf,sign

Author

Roger L. Bagula, Jun 14 2008

Status

approved