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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
1, 1, 1, 1, -6, 1, 1, 35, 35, 1, 1, -204, 1190, -204, 1, 1, 1189, 40426, 40426, 1189, 1, 1, -6930, 1373295, -8004348, 1373295, -6930, 1, 1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1, 1, -235416, 1584781276, -313786692648 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

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

4345425701134848, -9348927348636058624,...}.

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; this sequence at {1,-6,1}

LINKS

Table of n, a(n) for n=0..39.

FORMULA

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])].

EXAMPLE

{1},

{1, 1},

{1, -6, 1},

{1, 35, 35, 1},

{1, -204, 1190, -204, 1},

{1, 1189, 40426, 40426, 1189, 1},

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

{1, 40391, 46651605, 1584821667, 1584821667, 46651605, 40391, 1},

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

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

{1, -7997214, 1828836219245, -12301065925422312, 2435550753890846098, -14195430382223350260, 2435550753890846098, -12301065925422312, 1828836219245, -7997214, 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

A034801, A156599, A156600

Sequence in context: A176429 A157155 A022169 * A178232 A203338 A158116

Adjacent sequences:  A156598 A156599 A156600 * A156602 A156603 A156604

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Feb 11 2009

STATUS

approved

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Last modified August 20 04:17 EDT 2019. Contains 326139 sequences. (Running on oeis4.)