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 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). 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A194604 A078355 A107823 * A165969 A098373 A054793 Adjacent sequences:  A139812 A139813 A139814 * A139816 A139817 A139818 KEYWORD uned,tabf,sign AUTHOR Roger L. Bagula, Jun 14 2008 STATUS approved

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