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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 June 2 17:02 EDT 2020. Contains 334787 sequences. (Running on oeis4.)