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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
1, 1, 1, 1, -5, 1, 1, 24, 24, 1, 1, -115, 552, -115, 1, 1, 551, 12673, 12673, 551, 1, 1, -2640, 290928, -1394030, 290928, -2640, 1, 1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1, 1, -60605, 153318529, -16865038190, 80805530806 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are:

{1, 2, -3, 50, 324, 26450, -817452, 320045000, 47381970276, 88885777805000,

-63049671623224368,...}.

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

LINKS

Table of n, a(n) for n=0..40.

FORMULA

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])].

EXAMPLE

{1},

{1, 1},

{1, -5, 1},

{1, 24, 24, 1},

{1, -115, 552, -115, 1},

{1, 551, 12673, 12673, 551, 1},

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

{1, 12649, 6678672, 153331178, 153331178, 6678672, 12649, 1},

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

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

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

Sequence in context: A157154 A022168 A157212 * A152572 A203346 A176793

Adjacent sequences:  A156597 A156598 A156599 * A156601 A156602 A156603

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Feb 11 2009

STATUS

approved

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Last modified May 21 18:56 EDT 2019. Contains 323444 sequences. (Running on oeis4.)