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.

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A210553 Triangle of coefficients of polynomials v(n,x) jointly generated with A210552; see the Formula section.

Original entry on oeis.org

1, 2, 1, 3, 2, 2, 4, 3, 5, 3, 5, 4, 9, 8, 5, 6, 5, 14, 15, 15, 8, 7, 6, 20, 24, 31, 26, 13, 8, 7, 27, 35, 54, 57, 46, 21, 9, 8, 35, 48, 85, 104, 108, 80, 34, 10, 9, 44, 63, 125, 170, 209, 199, 139, 55, 11, 10, 54, 80, 175, 258, 360, 404, 366, 240, 89, 12, 11, 65, 99
Offset: 1

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Author

Clark Kimberling, Mar 22 2012

Keywords

Comments

Let T(n,k) denote the term in row n, column k.
T(n,n): A000045 (Fibonacci numbers)
T(n,n-1): A006367
T(n,n-2): A105423
T(n,1): 1,2,3,4,5,6,7,8,9,...
T(n,2): 1,2,3,4,5,6,7,8,9,...
T(n,3): A000096
T(n,4): A005563
T(n,5): A055831
T(n,6): A111694
Row sums: A000225
Alternating row sums: A052551
For a discussion and guide to related arrays, see A208510.

Examples

			First five rows:
1
2...1
3...2...2
4...3...5...3
5...4...9...8...5
First three polynomials v(n,x): 1, 2 + x , 3 + 2x + 2x^2.

Crossrefs

Cf. A210552, A208510.

Programs

  • Mathematica
    u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
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[%]   (* A210552 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210553 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

Formula

u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,
v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
where u(1,x)=1, v(1,x)=1.

A052634 Expansion of e.g.f. 1/((1-2*x^2)*(1-x)).

Original entry on oeis.org

1, 1, 6, 18, 168, 840, 10800, 75600, 1249920, 11249280, 228614400, 2514758400, 60833203200, 790831641600, 22230464256000, 333456963840000, 10691545632768000, 181756275757056000, 6549628300959744000
Offset: 0

Views

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

Links

Programs

  • Maple
    spec := [S,{S=Prod(Sequence(Prod(Z,Union(Z,Z))),Sequence(Z))},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
  • Mathematica
    With[{nn=20},CoefficientList[Series[1/((1-2x^2)(1-x)),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 13 2014 *)

Formula

E.g.f.: 1/(-1+2*x^2)/(-1+x).
Recurrence: {a(1)=1, a(0)=1, a(2)=6, (12+2*n^3+12*n^2+22*n)*a(n) +(-2*n^2-10*n-12)*a(n+1) +(-n-3)*a(n+2) +a(n+3)=0}.
a(n) = (-1+Sum(1/2*(1+2*_alpha)*_alpha^(-1-n), with _alpha=RootOf(-1+2*_Z^2)))*n! .
a(n) = n!*[2^floor(n/2+1)-1].
a(n)=n!*A052551(n). - R. J. Mathar, Jun 03 2022

A137865 Triangle read by rows, antidiagonals of an array formed by A000012 * A049310(transform).

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 0, 1, 1, 2, 3, 4, 3, 1, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 0, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 2, 3, 5, 12, 14, 11, 5, 1, 0
Offset: 0

Views

Author

Gary W. Adamson, Feb 18 2008

Keywords

Comments

Rows of the array tend to the Fibonacci sequence.
Row sums of the triangle = A052551: (1, 1, 3, 3, 7, 7, 15, 15, 31, 31, 63, 63, ...).

Examples

			First few rows of the array:
  1, 0, 1, 0, 1, 0,  1, ...
  1, 1, 1, 2, 1, 3,  1, ...
  1, 1, 2, 2, 4, 3,  7, ...
  1, 1, 2, 3, 4, 7,  7, ...
  1, 1, 2, 3, 5, 7, 12, ...
  1, 1, 2, 3, 5, 8, 12, ...
  ...
First few rows of the triangle:
  1;
  1, 0;
  1, 1, 1;
  1, 1, 1, 0;
  1, 1, 2, 2, 1;
  1, 1, 2, 2, 1, 0;
  1, 1, 2, 3, 4, 3,  1;
  1, 1, 2, 3, 4, 3,  1,  0;
  1, 1, 2, 3, 5, 7,  7,  4,  1;
  1, 1, 2, 3, 5, 7,  7,  4,  1, 0;
  1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1;
  1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 0;
  ...

Crossrefs

Cf. A049310, A052551.

Formula

Triangle read by rows, antidiagonals of an array formed by taking A000012 * A049310(transform); given A049310 unsigned.

A235501 Riordan array (1/(1-2*x^2), x/(1-x)).

Original entry on oeis.org

1, 0, 1, 2, 1, 1, 0, 3, 2, 1, 4, 3, 5, 3, 1, 0, 7, 8, 8, 4, 1, 8, 7, 15, 16, 12, 5, 1, 0, 15, 22, 31, 28, 17, 6, 1, 16, 15, 37, 53, 59, 45, 23, 7, 1, 0, 31, 52, 90, 112, 104, 68, 30, 8, 1, 32, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 0, 63, 114, 225
Offset: 0

Views

Author

Philippe Deléham, Jan 11 2014

Keywords

Comments

Row sums are A007179(n+1).

Examples

			Triangle begins (0<=k<=n):
1
0, 1
2, 1, 1
0, 3, 2, 1
4, 3, 5, 3, 1
0, 7, 8, 8, 4, 1
8, 7, 15, 16, 12, 5, 1
0, 15, 22, 31, 28, 17, 6, 1

Crossrefs

Cf. Columns: A077957, A052551, A077866.
Diagonals: A000012, A001477, A022856.
Cf. Similar sequences: A059260, A191582.

Formula

T(n,n)=1, T(2n,0)=2^n, T(2n+1,0)=0, T(n,k)=T(n-1,k-1)+T(n-1,k) for 0
T(n,k)=T(n-1,k)+T(n-1,k-1)+2*T(n-2,k)-T(n-3,k)-2*T(n-3,k-1), T(0,0)=1, T(1,0)=0, T(1,1)=1, T(n,k)=0 if k<0 or if k>n.
T(n,n)=1, T(n+1,n)=n, T(n+2,n)=n*(n+1)/2 + 2.
exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(3*x + 2*x^2/2! + x^3/3!) = 3*x + 8*x^2/2! + 16*x^3/3! + 28*x^4/4! + 45*x^5/5! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014
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