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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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 . [From Philippe Deléham, Mar 10 2009]

LINKS

Table of n, a(n) for n=0..31.

EXAMPLE

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.

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]

CROSSREFS

Cf. A135950, A022166, A157783, A157832.

Sequence in context: A286718 A204579 A113095 * A274615 A258895 A156890

Adjacent sequences:  A157781 A157782 A157783 * A157785 A157786 A157787

KEYWORD

sign,tabl

AUTHOR

Roger L. Bagula, Mar 06 2009

STATUS

approved

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Last modified January 18 23:05 EST 2019. Contains 319282 sequences. (Running on oeis4.)