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A156603 A q-factorial triangle sequence built of Cartan A_n polynomials as antidiagonals: p(x,n)=CartanAn(x,n): t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]]; 0
1, 1, 1, 1, 1, 2, 1, 1, 0, 6, 1, 1, -1, 0, 24, 1, 1, -2, 0, 0, 120, 1, 1, -3, -6, 0, 0, 720, 1, 1, -4, -24, 24, 0, 0, 5040, 1, 1, -5, -60, 504, 120, 0, 0, 40320, 1, 1, -6, -120, 3360, 27720, -720, 0, 0, 362880, 1, 1, -7, -210, 13800, 702240, -3991680, -5040, 0, 0, 3628800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Row sums are:

{1, 2, 4, 8, 25, 120, 713, 5038, 40881, 393116, 347905,...}.

LINKS

Table of n, a(n) for n=0..65.

FORMULA

p(x,n)=CartanAn(x,n):

t(n,k)=If[m == 0, n!, Product[p(m+1),k), {k, 1, n}]];

Out_(n,m)=antidiagonal(t(n,m)).

EXAMPLE

{1},

{1, 1},

{1, 1, 2},

{1, 1, 0, 6},

{1, 1, -1, 0, 24},

{1, 1, -2, 0, 0, 120},

{1, 1, -3, -6, 0, 0, 720},

{1, 1, -4, -24, 24, 0, 0, 5040},

{1, 1, -5, -60, 504, 120, 0, 0, 40320},

{1, 1, -6, -120, 3360, 27720, -720, 0, 0, 362880},

{1, 1, -7, -210, 13800, 702240, -3991680, -5040, 0, 0, 3628800}

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}]];

a = Table[Table[t[n, m], {n, 0, 10}], {m, 0, 10}];

b = Table[Table[a[[m, n - m + 1]], {m, n, 1, -1}], {n, 1, Length[a]}]'

Flatten[%]

CROSSREFS

A034801, A156599, A156600, A156601, A156602

Sequence in context: A229892 A064879 A173591 * A156612 A096801 A072407

Adjacent sequences:  A156600 A156601 A156602 * A156604 A156605 A156606

KEYWORD

sign,tabl,uned

AUTHOR

Roger L. Bagula, Feb 11 2009

STATUS

approved

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Last modified September 20 01:58 EDT 2019. Contains 327207 sequences. (Running on oeis4.)