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.

A136688 Triangle of polynomials F(x,n) = x*F(x,n-1) + 2*F(x,n-2).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 4, 0, 1, 4, 0, 6, 0, 1, 0, 12, 0, 8, 0, 1, 8, 0, 24, 0, 10, 0, 1, 0, 32, 0, 40, 0, 12, 0, 1, 16, 0, 80, 0, 60, 0, 14, 0, 1, 0, 80, 0, 160, 0, 84, 0, 16, 0, 1, 32, 0, 240, 0, 280, 0, 112, 0, 18, 0, 1, 0, 192, 0, 560, 0, 448, 0, 144, 0, 20, 0, 1
Offset: 1

Views

Author

Roger L. Bagula, Apr 06 2008

Keywords

Comments

Riordan array (1/(1-2*x^2), x/(1-2*x^2)). - Paul Barry, Jun 18 2008
Antidiagonal sums are 1,0,3,0,9,... with g.f. 1/(1-3*x^2). - Paul Barry, Jun 18 2008

Examples

			Triangle begins:
   1;
   0,   1;
   2,   0,   1;
   0,   4,   0,   1;
   4,   0,   6,   0,   1;
   0,  12,   0,   8,   0,   1;
   8,   0,  24,   0,  10,   0,   1;
   0,  32,   0,  40,   0,  12,   0,   1;
  16,   0,  80,   0,  60,   0,  14,   0,   1;
   0,  80,   0, 160,   0,  84,   0,  16,   0,   1;
  32,   0, 240,   0, 280,   0, 112,   0,  18,   0,   1;
  ...

Links

Crossrefs

Row sums give A001045.
Cf. A136689, A136705.

Programs

  • Maple
    A136688 := proc(n) option remember: if(n<=1)then return n: else return x*A136688(n-1)+2*A136688(n-2): fi: end:
seq(seq(coeff(A136688(n),x,m),m=0..n-1),n=1..10); # Nathaniel Johnston, Apr 27 2011
  • Mathematica
    s = 2; F[x_, n_]:= F[x, n]= If[n<2, n, x*F[x, n-1] + s*F[x, n-2]]; Table[CoefficientList[F[x, n], x], {n,12}]//Flatten
F[n_, x_, s_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^Binomial[j+1, 2]*x^(n-2*j-1) *s^j, {j, 0, Floor[(n-1)/2]}]; Table[CoefficientList[F[n,x,2,1], x], {n, 1, 10}]//Flatten (* G. C. Greubel, Dec 16 2019 *)
  • PARI
    T(n,k)=if((n-k)%2==0, 0, 2^((n-k-1)/2)*binomial((n+k-1)/2, (n-k-1)/2)) \\ Andrew Howroyd, Feb 11 2023
  • Sage
    def f(n,x,s,q): return sum( q_binomial(n-j-1, j, q)*q^binomial(j+1,2)*x^(n-2*j-1)*s^j for j in (0..floor((n-1)/2)))
def A136688_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(n,x,2,1) ).list()
[A136688_list(n) for n in (1..10)] # G. C. Greubel, Dec 16 2019

Formula

F(x,n) = x*F(x,n-1) + s*F(x,n-2), where F(x,0)=0, F(x,1)=1 and s=2.
F(x,n) = Sum_{j=0..floor((n-1)/2)} binomial(n-j-1, j)*x^(n-2*j-1)*2^j, for n >= 1. See the Mma program by G. C. Greubel. - Wolfdieter Lang, Feb 10 2023
From Andrew Howroyd, Feb 11 2023: (Start)
T(n,k) = 2^((n-k-1)/2)*binomial((n+k-1)/2, (n-k-1)/2) for k+1 == n (mod 2).
G.f.: x/(1 - y*x - 2*x^2). (End)

A136689 Triangular sequence of q-Fibonacci polynomials for s=3: F(x,n) = x*F(x,n-1) + s*F(x,n-2).

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 6, 0, 1, 9, 0, 9, 0, 1, 0, 27, 0, 12, 0, 1, 27, 0, 54, 0, 15, 0, 1, 0, 108, 0, 90, 0, 18, 0, 1, 81, 0, 270, 0, 135, 0, 21, 0, 1, 0, 405, 0, 540, 0, 189, 0, 24, 0, 1, 243, 0, 1215, 0, 945, 0, 252, 0, 27, 0, 1, 0, 1458, 0, 2835, 0, 1512, 0
Offset: 1

Views

Author

Roger L. Bagula, Apr 06 2008

Keywords

Comments

Row sums: 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, ... = A006130(n-1).

Examples

			Triangle begins:
    1;
    0,   1;
    3,   0,    1;
    0,   6,    0,   1;
    9,   0,    9,   0,   1;
    0,  27,    0,  12,   0,   1;
   27,   0,   54,   0,  15,   0,   1;
    0, 108,    0,  90,   0,  18,   0,  1;
   81,   0,  270,   0, 135,   0,  21,  0,  1;
    0, 405,    0, 540,   0, 189,   0, 24,  0, 1;
  243,   0, 1215,   0, 945,   0, 252,  0, 27, 0, 1;
  ...

Links

Crossrefs

Cf. A136688, A136705.

Programs

  • Maple
    A136689 := proc(n) option remember: if(n<=1)then return n: else return x*procname(n-1)+3*procname(n-2): fi: end:
seq(seq(coeff(A136689(n), x, m), m=0..n-1), n=1..10); # Nathaniel Johnston, Apr 27 2011
  • Mathematica
    s=2; F[x_, n_]:= F[x, n]= If[n<2, n, x*F[x, n-1] + s*F[x, n-2]]; Table[
CoefficientList[F[x, n], x], {n,10}]//Flatten
F[n_, x_, s_, q_]:= Sum[QBinomial[n-j-1, j, q]*q^Binomial[j+1, 2]*x^(n-2*j-1) *s^j, {j, 0, Floor[(n-1)/2]}]; Table[CoefficientList[F[n,x,3,1], x], {n, 1, 10}]//Flatten (* G. C. Greubel, Dec 16 2019 *)
  • Sage
    def f(n,x,s,q): return sum( q_binomial(n-j-1, j, q)*q^binomial(j+1,2)*x^(n-2*j-1)*s^j for j in (0..floor((n-1)/2)))
def A136689_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(n,x,3,1) ).list()
[A136689_list(n) for n in (1..10)] # G. C. Greubel, Dec 16 2019

Formula

F(x,n) = x*F(x,n-1) + s*F(x,n-2), where F(x,0)=0, F(x,1)=1 and s=3.

A141679 Triangle of coefficients of the inverse of A058071.

Original entry on oeis.org

1, -1, 1, -1, -1, 1, 0, -1, -1, 1, 0, 0, -1, -1, 1, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1
Offset: 1

Views

Author

Roger L. Bagula and Gary W. Adamson, Sep 07 2008

Keywords

Comments

The row sums are {1, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, ...}.
The inverse is a tridiagonal lower triangular matrix.

Examples

			{1},
{-1, 1},
{-1, -1, 1},
{0, -1, -1, 1},
{0, 0, -1, -1, 1},
{0, 0,0, -1, -1, 1},
{0, 0, 0, 0, -1, -1, 1},
{0, 0, 0, 0, 0, -1, -1, 1},
{0, 0, 0, 0, 0, 0, -1, -1, 1},
{0, 0, 0, 0, 0, 0, 0, -1, -1, 1},
{0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 1}

Links

Crossrefs

Cf. A058071.
As a sequence, quite similar to A136705. - N. J. A. Sloane, Dec 14 2014

Programs

  • Mathematica
    Clear[t, n, m, M] (*A058071*) t[n_, m_] = If[m <= n, Fibonacci[n - m + 1]*Fibonacci[m + 1], 0]; Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}]; Flatten[%]; M = Inverse[Table[Table[t[n, m], {m, 0, 10}], {n, 0, 10}]]; Table[Table[Fibonacci[n]*M[[n, m]], {m, 1, n}], {n, 1, 11}]; Flatten[%]

Formula

A058071(n,m) = if(m <= n, Fibonacci(n - m + 1)*Fibonacci(m + 1), 0), t(n,m) = Fibonacci(n)*Inverse(A058071(n,m)).

Extensions

Edited by N. J. A. Sloane, Jan 05 2009
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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