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A156600 A q-combination triangle sequence built of Cartan A_n polynomials: m=6;q=7; 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])]. 5

%I

%S 1,1,1,1,-5,1,1,24,24,1,1,-115,552,-115,1,1,551,12673,12673,551,1,1,

%T -2640,290928,-1394030,290928,-2640,1,1,12649,6678672,153331178,

%U 153331178,6678672,12649,1,1,-60605,153318529,-16865038190,80805530806

%N A q-combination triangle sequence built of Cartan A_n polynomials: m=6;q=7; 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, -3, 50, 324, 26450, -817452, 320045000, 47381970276, 88885777805000,

%C -63049671623224368,...}.

%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; this sequence at {1,-5,1}

%F m=6;q=7; 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, -5, 1},

%e {1, 24, 24, 1},

%e {1, -115, 552, -115, 1},

%e {1, 551, 12673, 12673, 551, 1},

%e {1, -2640, 290928, -1394030, 290928, -2640, 1},

%e {1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1},

%e {1, -60605, 153318529, -16865038190, 80805530806, -16865038190, 153318529, -60605, 1},

%e {1, 290376, 3519647496, 1855000882371, 42584368082256, 42584368082256, 1855000882371, 3519647496, 290376, 1},

%e {1, -1391275, 80798573880, -204033232083225, 22441881327136635, -107525529407696400, 22441881327136635, -204033232083225, 80798573880, -1391275, 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

%K sign,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Feb 11 2009

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Last modified November 13 12:45 EST 2019. Contains 329094 sequences. (Running on oeis4.)