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A128894
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Triangle read by rows, giving dimensions of exceptional groups with extension to E9 as a non-simple Lie algebra.
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3
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3, 8, 27, 14, 77, 273, 28, 300, 1925, 8918, 52, 1053, 12376, 100776, 627912, 78, 2430, 43758, 537966, 4969107, 36685506, 133, 7371, 238602, 5248750, 85709988, 1101296924, 11604306012, 190, 15504, 749360, 24732110, 605537790, 11619550320, 181746027600, 2386644625950
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OFFSET
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1,1
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COMMENTS
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Row sums are {3, 35, 364, 11171, 742169, 42238845, 12796807780, ...}.
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LINKS
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FORMULA
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Let p = {-4/3, -1, -2/3, 0, 1, 2, 4, 6, 8, 16} then g(p,k) = (3*p + 2*k + 5)*binomial(k + 2*p + 3, k)*binomial(k + 5*p/2 + 3, k)*binomial(k + 3*p + 4, k)/((3*p + 5)*binomial(k + p/2 + 1, k)*binomial(k + p + 1, k)); see the Mathematica program.
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EXAMPLE
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Triangle begins:
3;
8, 27;
14, 77, 273;
28, 300, 1925, 8918;
52, 1053, 12376, 100776, 627912;
78, 2430, 43758, 537966, 4969107, 36685506; ...
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MATHEMATICA
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p = {-4/3, -1, -2/3, 0, 1, 2, 4, 6, 8, 16};
g[p_, k_] := (3*p +2*k +5) *Binomial[k+2*p+3, k]*Binomial[k+5*p/2 +3, k]*Binomial[k+3*p+4, k]/((3*p + 5)*Binomial[k+p/2 +1, k]*Binomial[k+p+1, k]);
Table[Table[g[p[[n]], k], {k, 1, n}], {n, 1, Length[p]}]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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