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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 . [From Philippe Deléham, Mar 09 2009]

LINKS

Table of n, a(n) for n=0..34.

EXAMPLE

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 November 11 18:50 EST 2019. Contains 329031 sequences. (Running on oeis4.)