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 A157783 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (3^(i-1)-x) in row n, column k, 0 <= k <= n. 5
 1, 1, -1, 3, -4, 1, 27, -39, 13, -1, 729, -1080, 390, -40, 1, 59049, -88209, 32670, -3630, 121, -1, 14348907, -21493836, 8027019, -914760, 33033, -364, 1, 10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Row sums except n=0 are zero. Triangle T(n,k), 0 <= k <= n, read by rows given by [1,q-1,q^2,q^3-q,q^4,q^5-q^2,q^6,q^7-q^3,q^8,...] DELTA [ -1,0,-q,0,-q^2,0,-q^3,0,-q^4,0,...] (for q=3)=[1,2,9,24,81,234,729,2160,6561,...] DELTA [ -1,0,-3,0,-9,0,-27,0,-81,0,-243,0,...] where DELTA is the operator defined in A084938; see A122006 and A000244. - Philippe Deléham, Mar 09 2009 LINKS EXAMPLE Triangle begins 1; 1, -1; 3, -4, 1; 27, -39, 13, -1; 729, -1080, 390, -40, 1; 59049, -88209, 32670, -3630, 121, -1; 14348907, -21493836, 8027019, -914760, 33033, -364, 1; 10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093, -1; 22876792454961, -34309958505840, 12860351387820, -1481851188720, 55340738838, -677572560, 2688780, -3280, 1; Row n=3 is 27 - 39*x + 13*x^2 - x^3. MAPLE A157783 := proc(n, k) product( 3^(i-1)-x, i=1..n) ; coeftayl(%, x=0, k) ; end proc: # R. J. Mathar, Oct 15 2013 MATHEMATICA Clear[f, q, M, n, m]; q = 3; f[k_, m_] := If[k == m, q^(n - k), If[m == 1 && k < n, q^(n - k), If[k == n && m == 1, -(n-1), If[k == n && m > 1, 1, 0]]]]; M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}]; Table[M[n], {n, 1, 10}]; Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]]; a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]]; Flatten[a] CROSSREFS Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2). Sequence in context: A162498 A134049 A224069 * A123951 A123127 A167876 Adjacent sequences: A157780 A157781 A157782 * A157784 A157785 A157786 KEYWORD sign,tabl AUTHOR Roger L. Bagula, Mar 06 2009 STATUS approved

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Last modified January 27 01:46 EST 2023. Contains 359836 sequences. (Running on oeis4.)