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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.

%I A157783 #11 Jan 26 2020 20:57:25
%S A157783 1,1,-1,3,-4,1,27,-39,13,-1,729,-1080,390,-40,1,59049,-88209,32670,
%T A157783 -3630,121,-1,14348907,-21493836,8027019,-914760,33033,-364,1,
%U A157783 10460353203,-15683355351,5873190687,-674887059,24995817,-298389,1093
%N 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.
%C A157783 Row sums except n=0 are zero.
%C A157783 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
%e A157783 Triangle begins
%e A157783   1;
%e A157783   1, -1;
%e A157783   3, -4, 1;
%e A157783   27, -39, 13, -1;
%e A157783   729, -1080, 390, -40, 1;
%e A157783   59049, -88209, 32670, -3630, 121, -1;
%e A157783   14348907, -21493836, 8027019, -914760, 33033, -364, 1;
%e A157783   10460353203, -15683355351, 5873190687, -674887059, 24995817, -298389, 1093, -1;
%e A157783   22876792454961, -34309958505840, 12860351387820, -1481851188720, 55340738838, -677572560, 2688780, -3280, 1;
%e A157783 Row n=3 is 27 - 39*x + 13*x^2 - x^3.
%p A157783 A157783 := proc(n,k)
%p A157783     product( 3^(i-1)-x,i=1..n) ;
%p A157783     coeftayl(%,x=0,k) ;
%p A157783 end proc: # _R. J. Mathar_, Oct 15 2013
%t A157783 Clear[f, q, M, n, m];
%t A157783 q = 3;
%t A157783 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]]]];
%t A157783 M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
%t A157783 Table[M[n], {n, 1, 10}];
%t A157783 Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
%t A157783 a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
%t A157783 Flatten[a]
%Y A157783 Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2).
%K A157783 sign,tabl
%O A157783 0,4
%A A157783 _Roger L. Bagula_, Mar 06 2009