A139278 -id:A139278 - cp's OEIS frontend

A207823 Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

1, 4, 1, 15, 8, 1, 56, 46, 12, 1, 209, 232, 93, 16, 1, 780, 1091, 592, 156, 20, 1, 2911, 4912, 3366, 1200, 235, 24, 1, 10864, 21468, 17784, 8010, 2120, 330, 28, 1, 40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1, 151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1

Offset: 0

Philippe Deléham, Feb 20 2012

Riordan array (1/(1-4*x+x^2), x/(1-4*x+x^2)).
Subtriangle of the triangle given by (0, 4, -1/4, 1/4, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.
Unsigned version of triangles in A124029 and in A159764.
For 1<=k<=n, T(n,k) equals the number of (n-1)-length  words over {0,1,2,3,4} containing k-1 letters equal 4 and avoiding 01. - Milan Janjic, Dec 20 2016

			Triangle begins:
  1
  4, 1
  15, 8, 1
  56, 46, 12, 1
  209, 232, 93, 16, 1
  780, 1091, 592, 156, 20, 1
  2911, 4912, 3366, 1200, 235, 24, 1
  10864, 21468, 17784, 8010, 2120, 330, 28, 1
  40545, 91824, 89238, 48624, 16255, 3416, 441, 32, 1
  151316, 386373, 430992, 275724, 111524, 29589, 5152, 568, 36, 1
  ...
Triangle (0, 4, -1/4, 1/4, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins:
  1
  0, 1
  0, 4, 1
  0, 15, 8, 1
  0, 56, 46, 12, 1
  0, 209, 232, 93, 16, 1
  ...

Rigoberto Flórez, Leandro Junes, José L. Ramírez, Further Results on Paths in an n-Dimensional Cubic Lattice, J. Int. Seq. 21 (2018), #18.1.2.
Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Cf. Triangle of coefficients of Chebyshev's S(n,x+k) polynomials: A207824 (k = 5), A207823 (k = 4), A125662 (k = 3), A078812 (k = 2), A101950 (k = 1), A049310 (k = 0), A104562 (k = -1), A053122 (k = -2), A207815 (k = -3), A159764 (k = -4), A123967 (k = -5).

With[{n = 9}, DeleteCases[#, 0] & /@ CoefficientList[Series[1/(1 - 4 x + x^2 - y x), {x, 0, n}, {y, 0, n}], {x, y}]] // Flatten (* Michael De Vlieger, Apr 25 2018 *)

Recurrence: T(n,k) = 4*T(n-1,k) + T(n-1,k-1) - T(n-2,k).
Diagonal sums are 4^n = A000302(n).
Row sums are A004254(n+1).
G.f.: 1/(1-4*x+x^2-y*x)
T(n,n) = 1, T(n+1,n) = 4*n+4 = A008586(n+1), T(n+2,n) = (n+1)*(8n+15) = A139278(n+1).
T(n,0) = A001353(n+1).

Offset changed to 0 by Georg Fischer, Feb 18 2020

A139277 a(n) = n(8n+5).

Original entry on oeis.org

0, 13, 42, 87, 148, 225, 318, 427, 552, 693, 850, 1023, 1212, 1417, 1638, 1875, 2128, 2397, 2682, 2983, 3300, 3633, 3982, 4347, 4728, 5125, 5538, 5967, 6412, 6873, 7350, 7843, 8352, 8877, 9418, 9975, 10548, 11137, 11742, 12363, 13000

Offset: 0

Omar E. Pol, Apr 26 2008

Sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139273 in the same spiral.

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250.

Cf. A139271, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Table[n (8 n + 5), {n, 0, 50}] (* Bruno Berselli, Aug 22 2018 *)
LinearRecurrence[{3,-3,1},{0,13,42},50] (* Harvey P. Dale, Dec 04 2018 *)

a(n)=n*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 8*n^2 + 5*n.
Sequences of the form a(n) = 8*n^2 + c*n have generating functions x*{c+8 + (8-c)*x}/(1-x)^3 and recurrence a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - R. J. Mathar, May 12 2008
a(n) = 16*n + a(n-1) - 3 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010
Sum_{n>=1} 1/a(n) = (sqrt(2)-1)*Pi/10 - 4*log(2)/5 + sqrt(2)*log(sqrt(2)+1)/5 + 8/25. - Amiram Eldar, Mar 18 2022
E.g.f.: exp(x)*x*(13 + 8*x). - Elmo R. Oliveira, Dec 15 2024

A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

Original entry on oeis.org

1, 4, -1, 15, -8, 1, 56, -46, 12, -1, 209, -232, 93, -16, 1, 780, -1091, 592, -156, 20, -1, 2911, -4912, 3366, -1200, 235, -24, 1, 10864, -21468, 17784, -8010, 2120, -330, 28, -1, 40545, -91824, 89238, -48624, 16255, -3416, 441, -32, 1, 151316, -386373, 430992, -275724, 111524, -29589, 5152, -568, 36, -1

Offset: 0

Gary W. Adamson and Roger L. Bagula, Nov 01 2006

The matrices are {4} if n=1, {{4,-1},{-1,4}} if n=2, {{4,-1,0},{-1,4,-1},{0,-1,4}} if n=3 etc. The empty matrix at n=0 has an empty product (determinant) with assigned value =1.

Riordan array (1/(1-4*x+x^2), -x/(1-4*x+x^2)). - Philippe Deléham, Mar 04 2016

			Triangle begins as:
      1;
      4,     -1;
     15,     -8,     1;
     56,    -46,    12,    -1;
    209,   -232,    93,   -16,    1;
    780,  -1091,   592,  -156,   20,   -1;
   2911,  -4912,  3366, -1200,  235,  -24,  1;
  10864, -21468, 17784, -8010, 2120, -330, 28, -1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Joanne Dombrowski, Tridiagonal matrix representations of cyclic self-adjoint operators, Pacific J. Math. 114, no. 2 (1984), 325-334.

Cf. A000302, A001353, A001906, A004254, A007070, A123966.

Cf. A139278, A159764, A181983, A207823 (absolute values).

m:=12;
R:=PowerSeriesRing(Integers(), m+2);
A124029:= func< n,k | Coefficient(R!( Evaluate(ChebyshevU(n+1), (4-x)/2) ), k) >;
[A124029(n,k): k in [0..n], n in [0..m]]; // G. C. Greubel, Aug 20 2023

A123966x := proc(n,x)
    local A,r,c ;
    A := Matrix(1..n,1..n) ;
    for r from 1 to n do
    for c from 1 to n do
            A[r,c] :=0 ;
        if r = c then
            A[r,c] := A[r,c]+4 ;
        elif abs(r-c)= 1 then
            A[r,c] :=  A[r,c]-1 ;
        end if;
    end do:
    end do:
    (-1)^n*LinearAlgebra[CharacteristicPolynomial](A,x) ;
end proc;
A123966 := proc(n,k)
    coeftayl( A123966x(n,x),x=0,k) ;
end proc:
seq(seq(A123966(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Dec 06 2011

(* Matrix version*)
k = 4;
T[n_, m_, d_]:= If[n==m, k, If[n==m-1 || n==m+1, -1, 0]];
M[d_]:= Table[T[n, m, d], {n,d}, {m,d}];
Table[M[d], {d,10}]
Table[Det[M[d]], {d,10}]
Table[Det[M[d] - x*IdentityMatrix[d]], {d,10}]
Join[{M[1]}, Table[CoefficientList[Det[M[ d] - x*IdentityMatrix[d]], x], {d,10}]]//Flatten
(* Recursive Polynomial form*)
p[0, x]= 1; p[1, x]= (4-x); p[k_, x_]:= p[k, x]= (4-x)*p[k-1, x] - p[k -2, x];
Table[CoefficientList[p[n, x], x], {n, 0, 10}]//Flatten
(* Additional program *)
Table[CoefficientList[ChebyshevU[n, (4-x)/2], x], {n,0,12}]//Flatten (* G. C. Greubel, Aug 20 2023 *)

def A124029(n,k): return ( chebyshev_U(n, (4-x)/2) ).series(x, n+2).list()[k]
flatten([[A124029(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 20 2023

T(n, k) = [x^k]( p(n, x) ), where p(n, x) = (4-x)*p(n-1, x) - p(n-2, x), p(0, x) = 1, p(1, x) = 4-x.
From G. C. Greubel, Aug 20 2023: (Start)
T(n, k) = [x^k]( ChebyshevU(n, (4-x)/2) ).
Sum_{k=0..n} T(n, k) = A001906(n+1).
Sum_{k=0..n} (-1)^k*T(n, k) = A004254(n+1).
Sum_{k=0..floor(n/2)} T(n-k, k) = A007070(n).
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = A000302(n).
T(n, n) = (-1)^n.
T(n, n-1) = 4*A181983(n), n >= 1.
T(n, n-2) = (-1)^n*A139278(n-1), n >= 2.
T(n, 0) = A001353(n+1). (End)

cp's OEIS Frontend

A207823 Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

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A139277 a(n) = n(8n+5).

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A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

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cp's OEIS Frontend

A207823 Triangle of coefficients of Chebyshev's S(n,x+4) polynomials (exponents of x in increasing order).

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Mathematica

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Extensions

A139277 a(n) = n*(8*n+5).

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Programs

Mathematica

PARI

Formula

A124029 Triangle T(n,k) with the coefficient [x^k] of the characteristic polynomial of the following n X n triangular matrix: 4 on the main diagonal, -1 of the two adjacent subdiagonals, 0 otherwise.

Views

Author

Keywords

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Examples

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Programs

Magma

Maple

Mathematica

SageMath

Formula

A139277 a(n) = n(8n+5).