cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A123966 Triangle A124029 with the (0,0) entry replaced by 4.

Original entry on oeis.org

4, 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
Offset: 0

Views

Author

Gary W. Adamson and Roger L. Bagula, Oct 28 2006

Keywords

Comments

The entry for the empty matrix in row 0 and column 0 is replaced by 4 in comparison to the variant in A124029.

Examples

			4;
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;

Crossrefs

Cf. A123343, A159764.

Programs

  • Mathematica
    Clear[M, T, d, a, x]; T[n_, m_] = If[ n == m, 4, If[n == m - 1 || n == m + 1, -1, 0]]; M[d_] := Table[T[n, m], {n, 1, d}, {m, 1, d}]; Table[M[d], {d, 1, 10}]; Table[Det[M[d] - x*IdentityMatrix[d]], {d, 1, 10}]; a = Join[{{3}}, Table[CoefficientList[Det[M[d] - x*IdentityMatrix[d]], x], {d, 1, 10}]]; Flatten[a]

A159764 Riordan array (1/(1+4x+x^2), x/(1+4x+x^2)).

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
Offset: 0

Views

Author

Paul Barry, Apr 21 2009

Keywords

Comments

Row sums are (-1)^n*F(2n+2). Diagonal sums are (-1)^n*4^n. Inverse is A052179.
The positive matrix is (1/(1-4x+x^2), x/(1-4x+x^2)) with general term T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,2),0).
For another version, see A124029.
Triangle of coefficients of Chebyshev's S(n,x-4) polynomials (exponents of x in increasing order). - Philippe Deléham, Feb 22 2012
Subtriangle of 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, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 22 2012

Examples

			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;
Triangle (0, -4, 1/4, -1/4, 0, 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;

Links

Crossrefs

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

Programs

  • Mathematica
    CoefficientList[CoefficientList[Series[1/(1 + 4*x + x^2 - y*x), {x, 0, 10}, {y, 0, 10}], x], y]//Flatten (* G. C. Greubel, May 21 2018 *)
  • Sage
    @CachedFunction
def A159764(n,k):
    if n< 0: return 0
    if n==0: return 1 if k == 0 else 0
    return A159764(n-1,k-1)-A159764(n-2,k)-4*A159764(n-1,k)
for n in (0..9): [A159764(n,k) for k in (0..n)] # Peter Luschny, Nov 20 2012

Formula

Number triangle T(n,k) = if(k<=n, Gegenbauer_C(n-k,k+1,-2),0).
G.f.: 1/(1+4*x+x^2-y*x). - Philippe Deléham, Feb 22 2012
T(n,k) = (-4)*T(n-1,k) + T(n-1,k-1) - T(n-2,k). - Philippe Deléham, Feb 22 2012

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

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

Views

Author

Philippe Deléham, Feb 20 2012

Keywords

Comments

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

Examples

			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
  ...

Links

Crossrefs

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).

Programs

  • Mathematica
    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 *)

Formula

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).

Extensions

Offset changed to 0 by Georg Fischer, Feb 18 2020
Showing 1-3 of 3 results.

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