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A156599 A q-combination triangle sequence built of Cartan A_n polynomials: b(n,k,m) = if(n=0, 1, t(n, m)/(t(k, m)*t(n-k, m))), where m=5, p(x,n) = CartanAn(x,n), and t(n,k) = Product_{k=1..n} p(m+1, k). 5
1, 1, 1, 1, -4, 1, 1, 15, 15, 1, 1, -56, 210, -56, 1, 1, 209, 2926, 2926, 209, 1, 1, -780, 40755, -152152, 40755, -780, 1, 1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1, 1, -10864, 7906276, -411126352, 1534382278, -411126352, 7906276, -10864, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Row sums are: {1, 2, -2, 32, 100, 6272, -72200, 16959488, 727920400, 638287290368, -102236420180000, ...}.

LINKS

G. C. Greubel, Rows n = 0..20 of triangle, flattened

FORMULA

With m=5, p(x,n) = CartanAn(x,n), and t(n,k) = Product_{k=1..n} p(m+1, k) then b(n,k,m) = if(n=0, 1, t(n, m)/(t(k, m)*t(n-k, m))).

EXAMPLE

Triangle begins:

  1;

  1,    1;

  1,   -4,      1;

  1,   15,     15,       1;

  1,  -56,    210,     -56,       1;

  1,  209,   2926,    2926,     209,      1;

  1, -780,  40755, -152152,   40755,   -780,    1;

  1, 2911, 567645, 7909187, 7909187, 567645, 2911, 1;

MATHEMATICA

r:= 5; T[n_, k_, d_]:= If[n==k, 2, If[n==k-1 || n==k+1, -1, 0]];

M[d_]:= Table[T[n, j, d], {n, 1, d}, {j, 1, d}];

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

a0:= Table[p[x, n], {n, 0, 10}] /. x -> r+1;

t[n_]:= Product[a0[[k]], {k, 1, n}];

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

Table[b[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 23 2019 *)

CROSSREFS

Cf. A034801.

Sequence in context: A157211 A176428 A116469 * A155826 A010320 A152571

Adjacent sequences:  A156596 A156597 A156598 * A156600 A156601 A156602

KEYWORD

sign,tabl,less

AUTHOR

Roger L. Bagula, Feb 11 2009

EXTENSIONS

Edited by G. C. Greubel, May 23 2019

STATUS

approved

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Last modified October 20 20:24 EDT 2019. Contains 328273 sequences. (Running on oeis4.)