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 A129766 Triangular array read by rows, made up of traditional exceptional groups plus A1: as A1,G2,F4,E6,E7,E8 as m(i) exponents as in A005556, A005763, A005776. 3
 1, 1, 5, 1, 5, 7, 11, 1, 4, 5, 7, 8, 11, 1, 5, 7, 9, 11, 13, 17, 1, 7, 11, 13, 17, 19, 23, 29 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Extra condition of group dimension: b[n] = a[n] + 1 ; DimGroup = Apply[Plus, b[n]]; Table[Apply[Plus, b[n]], {n, 0, 5}] {3, 14, 52, 78, 133, 248} Extra condition of Betti sum: Table[Apply[Plus, CoefficientList[Expand[Product[(1 + t^(2*a[i][[n]] + 1)), {n, 1,Length[a[i]]}]], t]], {i, 0, 5}] {2, 4, 16, 64, 128, 256} These exponents are necessary to the PoincarĂ© polynomials for these exceptional groups. LINKS Armand Borel, Essays in History of Lie Groups and Algebraic Groups gives G2 PoincarĂ© polynomial, History of Mathematics, V. 21. EXAMPLE 1; 1,5; 1,5,7,11; 1,4,5,7,8,11; 1,5,7,9,11,13,17; 1,7,11,13,17,19,23,29; MATHEMATICA a[0] = {1}; a[1] = {1, 5}; a[2] = {1, 5, 7, 11}; a[3] = {1, 4, 5, 7, 8, 11}; a[4] = {1, 5, 7, 9, 11, 13, 17}; a[5] = {1, 7, 11, 13, 17, 19, 23, 29}; Flatten[Table[a[n], {n, 0, 5}]] CROSSREFS Cf. A005556, A005763, A005776. Sequence in context: A011093 A062176 A129769 * A120283 A103986 A196404 Adjacent sequences:  A129763 A129764 A129765 * A129767 A129768 A129769 KEYWORD nonn,fini,full,tabf,uned AUTHOR Roger L. Bagula, May 16 2007 STATUS approved

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Last modified October 23 01:49 EDT 2018. Contains 316518 sequences. (Running on oeis4.)