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.

1, 1, -1, 4, -5, 1, 64, -84, 21, -1, 4096, -5440, 1428, -85, 1, 1048576, -1396736, 371008, -23188, 341, -1, 1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1, 4398046511104, -5863704100864, 1563272675328, -99158478848

Offset: 0

Author

Roger L. Bagula, Mar 06 2009

Keywords

• sign
• tabl

Comments

Row sums except n=0 are zero.

The matrix inverses seem to be related to the Gaussian q-form combinations.

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

Examples

			Triangle begins
  1;
  1, -1;
  4, -5, 1;
  64, -84, 21, -1;
  4096, -5440, 1428, -85, 1;
  1048576, -1396736, 371008, -23188, 341, -1;
  1073741824, -1431306240, 381308928, -24115520, 372372, -1365, 1;
  4398046511104, -5863704100864, 1563272675328, -99158478848, 1549351232, -5963412, 5461, -1;
  72057594037927936, -96075326035066880, 25618523216674816, -1626175790120960, 25483729063936, -99253893440, 95436436, -21845, 1;
Row n=3 represents 64 - 84*x + 21*x^2 - x^3.
		

Crossrefs

Cf. A135950, A022166, A157783, A157832.

Programs

Maple

A157784 := proc(n,k)
    product( 4^(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 = 4;
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]