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A156601 A q-combination triangle sequence built of Cartan A_n polynomials: m=7;q=8; 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])]. 2

%I

%S 1,1,1,1,-6,1,1,35,35,1,1,-204,1190,-204,1,1,1189,40426,40426,1189,1,

%T 1,-6930,1373295,-8004348,1373295,-6930,1,1,40391,46651605,1584821667,

%U 1584821667,46651605,40391,1,1,-235416,1584781276,-313786692648

%N A q-combination triangle sequence built of Cartan A_n polynomials: m=7;q=8; 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])].

%C Row sums are:

%C {1, 2, -4, 72, 784, 83232, -5271616, 3263027328, 1204479910144,

%C 4345425701134848, -9348927348636058624,...}.

%C I get as m levels:

%C m=0;Binomial

%C m=1;Indeterminant

%C m=2;Indeterminant

%C m=3;signed Binomial at {1.-2,1}

%C m=4;A034801 at {1,-3,1}

%C m=5;A156599 at {1,-4,1}

%C m=6;A156600 at {1,-5,1}

%C m=7; this sequence at {1,-6,1}

%F m=7;q=8; 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])].

%e {1},

%e {1, 1},

%e {1, -6, 1},

%e {1, 35, 35, 1},

%e {1, -204, 1190, -204, 1},

%e {1, 1189, 40426, 40426, 1189, 1},

%e {1, -6930, 1373295, -8004348, 1373295, -6930, 1},

%e {1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1},

%e {1, -235416, 1584781276, -313786692648, 1828884203718, -313786692648, 1584781276, -235416, 1},

%e {1, 1372105, 53835911780, 62128180363028, 2110530832920510, 2110530832920510, 62128180363028, 53835911780, 1372105, 1},

%e {1, -7997214, 1828836219245, -12301065925422312, 2435550753890846098, -14195430382223350260, 2435550753890846098, -12301065925422312, 1828836219245, -7997214, 1}

%t Clear[t, n, m, i, k, a, b, T, M, p];

%t T[n_, m_, d_] := If[ n == m, 2, If[n == m - 1 || n == m + 1, -1, 0]];

%t M[d_] := Table[T[n, m, d], {n, 1, d}, {m, 1, d}];

%t p[x_, n_] := If[n == 0, 1, CharacteristicPolynomial[M[n], x]];

%t a0 = Table[p[x, n], {n, 0, 20}] /. x -> m + 1;

%t t[n_, m_] = If[m == 0, n!, Product[a0[[k]], {k, 1, n}]];

%t b[n_, k_, m_] = If[n == 0, 1, t[n, m]/(t[k, m]*t[n - k, m])];

%t Table[Flatten[Table[Table[b[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 15}]

%Y A034801, A156599, A156600

%K sign,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Feb 11 2009

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Last modified October 17 19:24 EDT 2019. Contains 328127 sequences. (Running on oeis4.)