A122006 -id:A122006 - cp's OEIS frontend

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.

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

Offset: 0

Roger L. Bagula, Mar 06 2009

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. - Philippe Deléham, Mar 09 2009

			Triangle begins
  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.

Cf. A157832, A135950, A022166, A047656 (column k=1), A003462 (subdiagonal k=n-1), A203243 (subdiagonal k=n-2).

A157783 := proc(n,k)
    product( 3^(i-1)-x,i=1..n) ;
    coeftayl(%,x=0,k) ;
end proc: # R. J. Mathar, Oct 15 2013

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]

A167993 Expansion of x^2/((3x-1)(3*x^2-1)).

Original entry on oeis.org

0, 0, 1, 3, 12, 36, 117, 351, 1080, 3240, 9801, 29403, 88452, 265356, 796797, 2390391, 7173360, 21520080, 64566801, 193700403, 581120892, 1743362676, 5230147077, 15690441231, 47071500840, 141214502520, 423644039001, 1270932117003, 3812797945332, 11438393835996

Offset: 0

Paul Curtz, Nov 16 2009

The terms satisfy a(n) = 3*a(n-1) +3*a(n-2) -9*a(n-3), so they follow the pattern a(n) = p*a(n-1) +q*a(n-2) -p*q*a(n-3) with p=q=3. This could be called the principal sequence for that recurrence because we have set all but one of the initial terms to zero. [p=q=1 leads to the principal sequence A004526. p=q=2 leads essentially to A032085. The common feature is that the denominator of the generating function does not have a root at x=1, so the sequences of higher order successive differences have the same recurrence as the original sequence. See A135094, A010036, A006516.]

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (3,3,-9).

Cf. A138587, A107767 (partial sums).

CoefficientList[Series[x^2/((3*x - 1)*(3*x^2 - 1)), {x, 0, 50}], x] (* G. C. Greubel, Jul 03 2016 *)
LinearRecurrence[{3,3,-9},{0,0,1},30] (* Harvey P. Dale, Nov 05 2017 *)

Vec(x^2/((3*x-1)*(3*x^2-1))+O(x^99)) \\ Charles R Greathouse IV, Jun 29 2011

a(2*n+1) = 3*a(2*n).
a(2*n) = A122006(2*n)/2.
a(n) = 3*a(n-1) + 3*a(n-4) - 9*a(n-3).
a(n+1) - a(n) = A122006(n).
a(n) = (3^n - A108411(n+1))/6.
G.f.: x^2/((3*x-1)*(3*x^2-1)).
From Colin Barker, Sep 23 2016: (Start)
a(n) = 3^(n-1)/2-3^(n/2-1)/2 for n even.
a(n) = 3^(n-1)/2-3^(n/2-1/2)/2 for n odd.
(End)

Formulae corrected by Johannes W. Meijer, Jun 28 2011

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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.

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A167993 Expansion of x^2/((3x-1)(3*x^2-1)).

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cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

A167993 Expansion of x^2/((3*x-1)*(3*x^2-1)).

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Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A167993 Expansion of x^2/((3x-1)(3*x^2-1)).