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A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.

%I A157784 #11 Jan 26 2020 20:57:10
%S A157784 1,1,-1,4,-5,1,64,-84,21,-1,4096,-5440,1428,-85,1,1048576,-1396736,
%T A157784 371008,-23188,341,-1,1073741824,-1431306240,381308928,-24115520,
%U A157784 372372,-1365,1,4398046511104,-5863704100864,1563272675328,-99158478848
%N A157784 Triangle read by rows: the coefficient [x^k] of the polynomial Product_{i=1..n} (4^(i-1)-x), in row n and column 0 <= k <= n.
%C A157784 Row sums except n=0 are zero.
%C A157784 The matrix inverses seem to be related to the Gaussian q-form combinations.
%C A157784 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=4)=[1,3,16,60,256,1008,4096,16320,65536,261888,...] DELTA [ -1,0,-4,0,-16,0,-64,0,-256,0,-1024,0,...] where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 10 2009
%e A157784 Triangle begins
%e A157784   1;
%e A157784   1, -1;
%e A157784   4, -5, 1;
%e A157784   64, -84, 21, -1;
%e A157784   4096, -5440, 1428, -85, 1;
%e A157784   1048576, -1396736, 371008, -23188, 341, -1;
%e A157784   1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1;
%e A157784   4398046511104, -5863704100864, 1563272675328, -99158478848, 1549351232, -5963412, 5461, -1;
%e A157784   72057594037927936, -96075326035066880, 25618523216674816, -1626175790120960, 25483729063936, -99253893440, 95436436, -21845, 1;
%e A157784 Row n=3 represents 64 - 84*x + 21*x^2 - x^3.
%p A157784 A157784 := proc(n,k)
%p A157784     product( 4^(i-1)-x,i=1..n) ;
%p A157784     coeftayl(%,x=0,k) ;
%p A157784 end proc: # _R. J. Mathar_, Oct 15 2013
%t A157784 Clear[f, q, M, n, m];
%t A157784 q = 4;
%t A157784 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 A157784 M[n_] := Table[f[k, m], {k, 1, n}, {m, 1, n}];
%t A157784 Table[M[n], {n, 1, 10}];
%t A157784 Join[{1}, Table[Expand[CharacteristicPolynomial[M[n], x]], {n, 1, 7}]];
%t A157784 a = Join[{{ 1}}, Table[CoefficientList[CharacteristicPolynomial[M[n], x], x], {n, 1, 7}]];
%t A157784 Flatten[a]
%Y A157784 Cf. A135950, A022166, A157783, A157832.
%K A157784 sign,tabl
%O A157784 0,4
%A A157784 _Roger L. Bagula_, Mar 06 2009