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 A156602 A q-combination triangle sequence built of Cartan A_n polynomials: m=8;q=9; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])]. 3
 1, 1, 1, 1, -7, 1, 1, 48, 48, 1, 1, -329, 2256, -329, 1, 1, 2255, 105985, 105985, 2255, 1, 1, -15456, 4979040, -34127170, 4979040, -15456, 1, 1, 105937, 233908896, 10988845010, 10988845010, 233908896, 105937, 1, 1, -726103, 10988739073 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, -5, 98, 1600, 216482, -24200000, 22445719688, 17197609000000, 109329711296575112, -574146991795520000000,...}. I get as m levels: m=0;Binomial m=1;Indeterminant m=2;Indeterminant m=3;signed Binomial at {1.-2,1} m=4;A034801 at {1,-3,1} m=5;A156599 at {1,-4,1} m=6;A156600 at {1,-5,1} m=7;A156601 at {1,-6,1} m=8; this sequence at {1,-7,1}. These sequences are important because they relate Cartan quantum orthogonal group theory ( Lie algebra)to a set of combinatorial triangle sequences. As A_1 or SU(2) is associated to weak and electromagnetic theory and A_2 or SU(3) to strong nuclear force and A_4 or SU(5) to the standard model of physics, a q particle modeled on these combinations would be very important to physics. LINKS FORMULA m=8;q=9; p(x,n)=CartanAn(x,n). t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]]; b(n,k,m)=If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])]. EXAMPLE {1}, {1, 1}, {1, -7, 1}, {1, 48, 48, 1}, {1, -329, 2256, -329, 1}, {1, 2255, 105985, 105985, 2255, 1}, {1, -15456, 4979040, -34127170, 4979040, -15456, 1}, {1, 105937, 233908896, 10988845010, 10988845010, 233908896, 105937, 1}, {1, -726103, 10988739073, -3538373981506, 24252380937070, -3538373981506, 10988739073, -726103, 1}, {1, 4976784, 516236827536, 1139345433305859, 53524993973177376, 53524993973177376, 1139345433305859, 516236827536, 4976784, 1}, {1, -34111385, 24252142155120, -366865691151231195, 118129637457410270835, -809672583832254166752, 118129637457410270835, -366865691151231195, 24252142155120, -34111385, 1} MATHEMATICA Clear[t, n, m, i, k, a, b, T, M, p]; T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}]; p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]]; a0 = Table[p[x, n], {n, 0, 20}] /. x -> m + 1; t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]]; b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])]; Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}] CROSSREFS Sequence in context: A157156 A022170 A178658 * A203389 A174689 A172345 Adjacent sequences:  A156599 A156600 A156601 * A156603 A156604 A156605 KEYWORD sign,tabl,uned AUTHOR Roger L. Bagula, Feb 11 2009 STATUS approved

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Last modified May 27 21:12 EDT 2020. Contains 334671 sequences. (Running on oeis4.)