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 1-4 of 4 results.

A091004 Expansion of x*(1-x)/((1-2*x)*(1+3*x)).

Original entry on oeis.org

0, 1, -2, 8, -20, 68, -188, 596, -1724, 5300, -15644, 47444, -141308, 425972, -1273820, 3829652, -11472572, 34450484, -103285916, 309988820, -929704316, 2789637236, -8367863132, 25105686548, -75312865340, 225946984628, -677824176668, 2033506084436
Offset: 0

Views

Author

Paul Barry, Dec 13 2003

Keywords

Comments

Inverse binomial transform of A091001.

Links

Crossrefs

Cf. A091001, A091003, A091005, A112555.

Programs

  • GAP
    Concatenation([0], List([1..30], n -> (3*2^n - 8*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019
  • Magma
    [0] cat [(3*2^n - 8*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019
  • Mathematica
    CoefficientList[Series[x(1-x)/((1-2x)(1+3x)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 17 2017 *)
Join[{0}, LinearRecurrence[{-1, 6}, {1, -2}, 30]] (* G. C. Greubel, Feb 01 2019 *)
  • PARI
    vector(30, n, n--; (3*2^n - 8*(-3)^n + 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019
  • Sage
    [0] + [(3*2^n - 8*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

Formula

G.f.: x*(1-x)/((1-2*x)*(1+3*x)).
a(n) = (3*2^n - 8*(-3)^n + 5*0^n)/30.
2^n = A091003(n) + 3*a(n) + 6*A091005(n).
a(n+1) = Sum_{k=0..n} A112555(n,k)*(-3)^k. - Philippe Deléham, Sep 11 2009
E.g.f.: (3*exp(2*x) - 8*exp(-3*x) + 5)/30. - G. C. Greubel, Feb 01 2019

A091005 Expansion of x^2/((1-2*x)*(1+3*x)).

Original entry on oeis.org

0, 0, 1, -1, 7, -13, 55, -133, 463, -1261, 4039, -11605, 35839, -105469, 320503, -953317, 2876335, -8596237, 25854247, -77431669, 232557151, -697147165, 2092490071, -6275373061, 18830313487, -56482551853, 169464432775, -508359743893, 1525146340543
Offset: 0

Views

Author

Paul Barry, Dec 13 2003

Keywords

Comments

Inverse binomial transform of A091002.

Links

Crossrefs

Cf. A015441.

Programs

  • GAP
    Concatenation([0], List([1..30], n -> (3*2^n + 2*(-3)^n)/30)); # G. C. Greubel, Feb 01 2019
  • Magma
    [0] cat [(3*2^n + 2*(-3)^n)/30: n in [1..30]]; // G. C. Greubel, Feb 01 2019
  • Mathematica
    a[n_]:=(MatrixPower[{{1,4},{1,-2}},n].{{1},{1}})[[2,1]]; Table[a[n],{n,0,40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
Join[{0, 0}, LinearRecurrence[{-1, 6}, {1, -1}, 30]] (* G. C. Greubel, Feb 01 2019 *)
CoefficientList[Series[x^2/((1-2x)(1+3x)),{x,0,30}],x] (* Harvey P. Dale, Apr 30 2022 *)
  • PARI
    vector(30, n, n--; (3*2^n + 2*(-3)^n - 5*0^n)/30) \\ G. C. Greubel, Feb 01 2019
  • Sage
    [0] + [(3*2^n + 2*(-3)^n)/30 for n in (1..30)] # G. C. Greubel, Feb 01 2019

Formula

2^n = A091003(n) + 3*A091004(n) + 6*a(n).
a(n) = (3*2^n + 2*(-3)^n - 5*0^n)/30.
E.g.f.: (3*exp(2*x) + 2*exp(-3*x) - 5)/30. - G. C. Greubel, Feb 01 2019

A208343 Triangle of coefficients of polynomials v(n,x) jointly generated with A208342; see the Formula section.

Original entry on oeis.org

1, 0, 2, 0, 1, 3, 0, 1, 2, 5, 0, 1, 2, 5, 8, 0, 1, 2, 6, 10, 13, 0, 1, 2, 7, 13, 20, 21, 0, 1, 2, 8, 16, 29, 38, 34, 0, 1, 2, 9, 19, 39, 60, 71, 55, 0, 1, 2, 10, 22, 50, 86, 122, 130, 89, 0, 1, 2, 11, 25, 62, 116, 187, 241, 235, 144, 0, 1, 2, 12, 28, 75, 150, 267, 392, 468
Offset: 1

Views

Author

Clark Kimberling, Feb 25 2012

Keywords

Comments

u(n,n) = A000045(n+1) (Fibonacci numbers).
n-th row sum: 2^(n-1)
As triangle T(n,k) with 0 <= k <= n, it is (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 26 2012

Examples

			First five rows:
  1;
  0, 2;
  0, 1, 3;
  0, 1, 2, 5;
  0, 1, 2, 5, 8;
First five polynomials v(n,x):
  1
     2x
      x + 3x^2
      x + 2x^2 + 5x^3
      x + 2x^2 + 5x^3 + 8x^4.

Crossrefs

Cf. A208343.
Cf. A084938, A000045, A000079.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 13;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]  (* A208342 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A208343 *)

Formula

u(n,x) = u(n-1,x) + x*v(n-1,x),
v(n,x) = x*u(n-1,x) + x*v(n-1,x),
where u(1,x)=1, v(1,x)=1.
From Philippe Deléham, Feb 26 2012: (Start)
As triangle T(n,k) with 0 <= k <= n:
T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k > n or if k < 0.
G.f.: (1-(1-y)*x)/(1-(1+y)*x+y*(1-y)*x^2).
Sum_{k=0..n} T(n,k)*x^k = (-1)*A091003(n+1), A152166(n), A000007(n), A000079(n), A055099(n), A152224(n) for x = -2, -1, 0, 1, 2, 3 respectively.
Sum_{k=0..n} T(n,k)*x^(n-k) = A087205(n), A140165(n+1), A016116(n+1), A000045(n+2), A000079(n), A122367(n), A006012(n), A052961(n), A154626(n) for x = -3, -2, -1, 0, 1, 2, 3, 4 respectively. (End)
T(n,k) = A208748(n,k)/2^k. - Philippe Deléham, Mar 05 2012

A140725 Inverse binomial transform of (0 followed by A037481).

Original entry on oeis.org

0, 1, 4, 10, 34, 94, 298, 862, 2650, 7822, 23722, 70654, 212986, 636910, 1914826, 5736286, 17225242, 51642958, 154994410, 464852158, 1394818618, 4183931566, 12552843274, 37656432670, 112973492314, 338912088334, 1016753042218
Offset: 0

Views

Author

Paul Curtz, Jul 12 2008

Keywords

Comments

From Sean A. Irvine, Jun 07 2025: (Start)
For n>=1, the number of walks of length n-1 starting at vertex 1 (or, by symmetry, vertex 4) in the graph K_{1,1,3}:
1---2
/|\ /
0 | X
\|/ \
4---3. (End)

Links

Crossrefs

Cf. A083421 (bin. transform of (0 followed by A037481)).

Programs

  • Mathematica
    Join[{0},LinearRecurrence[{1,6},{1,4},26]] (* or *) a[0]=0;a[n_]:= ((-2)^n+4*3^n)/10;Array[a,27,0] (* James C. McMahon, Jul 13 2025 *)

Formula

a(n)= (-1)^n*A091003(n), n>0.
a(n+1)-3*a(n) = (-1)^(n+1)*A000079(n-1), n>0.
|a(n+1)-3*a(n)| = A011782(n).
From R. J. Mathar, Jul 14 2008: (Start)
O.g.f.: (1+3*x)*x / ((1+2*x)*(1-3*x)).
a(n) = ((-2)^n+4*3^n)/10, n>0. (End)
a(n) = a(n-1)+6*a(n-2) for n>2, a(0)=0, a(1)=1, a(2)=4. - Philippe Deléham, Nov 17 2013
a(n) + a(n+1) = A140796(n). - Philippe Deléham, Nov 17 2013
a(n+1) = sum_{k=0..n} A108561(n,k)*(-3)^k. - Philippe Deléham, Nov 17 2013

Extensions

Edited and extended by R. J. Mathar, Jul 14 2008
Showing 1-4 of 4 results.

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