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.

A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).

Original entry on oeis.org

1, 2, 6, 20, 70, 250, 900, 3250, 11750, 42500, 153750, 556250, 2012500, 7281250, 26343750, 95312500, 344843750, 1247656250, 4514062500, 16332031250, 59089843750, 213789062500, 773496093750, 2798535156250, 10125195312500
Offset: 0

Views

Author

Paul Barry, Nov 11 2008

Keywords

Comments

Row sums of A147747. Binomial transform of A061646.
Counts all paths of length (2*n), n>=0, starting at the initial node on the path graph P_9, see the Maple program. - Johannes W. Meijer, May 29 2010
From L. Edson Jeffery, Apr 19 2011: (Start)
For the 5 X 5 unit-primitive matrix (see [Jeffery])
A_(10,1) = [0,1,0,0,0; 1,0,1,0,0; 0,1,0,1,0; 0,0,1,0,1; 0,0,0,2,0],
a(n) = (Trace([A_(10,1)]^(2*n)))/5. (See also A189315.) (End)

Links

Crossrefs

Cf. A000984, A030191, A061646, A081567, A147746, A147747, A178381, A189315.

Programs

  • Maple
    with(GraphTheory): G:=PathGraph(9): A:= AdjacencyMatrix(G): nmax:=24; n2:=nmax*2: for n from 0 to n2 do B(n):=A^n; a(n):= add(B(n)[1,k], k=1..9); od: seq(a(2*n), n=0..nmax); # Johannes W. Meijer, May 29 2010
  • Mathematica
    (1 - 3x + x^2)/(1 - 5x + 5x^2) + O[x]^25 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 05 2016 *)

Formula

G.f.: (1-3*x+x^2)/(1-5*x+5*x^2).
a(n) = 5*a(n-1) - 5*a(n-2) for n > 2, a(0)=1, a(1)=2, a(2)=6. - Philippe Deléham, Nov 13 2008
For n >= 1: a(n) = (2/5)*((5-sqrt(5))/2)^n + (2/5)*((5+sqrt(5))/2)^n. - Richard Choulet, Nov 14 2008
G.f.: 1/(1-2x/(1-x/(1-x/(1-x)))) (hence sequence approximates A000984 in first few terms). - Paul Barry, Aug 05 2009
a(n) = (1/5)*Sum_{k=1..5} (x_k)^(2*n), x_k=2*cos((2*k-1)*Pi/10). - L. Edson Jeffery, Apr 19 2011
From R. J. Mathar, Apr 20 2011: (Start)
a(n) = A030191(n) - 3*A030191(n-1) + A030191(n-2).
a(n) = 2*A081567(n-1), n > 0. (End)
a(n) = Sum_{k=0..n} A147746(n,k)*2^k. - Philippe Deléham, Oct 30 2011
E.g.f.: (1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2))/5. - Stefano Spezia, Jul 09 2024

A147746 Riordan array (1, x(1-2x)/(1-3x+x^2)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 5, 3, 1, 0, 13, 14, 9, 4, 1, 0, 34, 40, 28, 14, 5, 1, 0, 89, 114, 87, 48, 20, 6, 1, 0, 233, 323, 267, 161, 75, 27, 7, 1, 0, 610, 910, 809, 528, 270, 110, 35, 8, 1
Offset: 0

Views

Author

Paul Barry, Nov 11 2008

Keywords

Comments

Triangle [0,1,1,1,0,0,0,....] DELTA [1,0,0,0,...] with Deléham DELTA as in A084938.
Note that 1/(1-x/(1-x/(1-x))) = (1-2x)/(1-3x+x^2). Row sums are A124302.
A147746*A007318 = A147747.

Examples

			Triangle begins
  1;
  0,   1;
  0,   1,   1;
  0,   2,   2,   1;
  0,   5,   5,   3,   1;
  0,  13,  14,   9,   4,   1;
  0,  34,  40,  28,  14,   5,   1;
  0,  89, 114,  87,  48,  20,   6,   1;
  ...

Programs

  • Mathematica
    (* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, # (1-2#)/(1-3#+#^2)&, 10] // Flatten (* Jean-François Alcover, Jul 19 2019 *)

Formula

Sum_{k=0..n} T(n,k)*2^k = A147748(n). - Philippe Deléham, Oct 30 2011
Sum_{k=0..n} T(n,k)*(-1)^(n-k) = A215936(n). - Philippe Deléham, Aug 30 2012
G.f.: (1 - 3*x + x^2)/(1 - 3*x + x^2 - x*y + 2*x^2*y). - R. J. Mathar, Aug 11 2015

A147750 Riordan array ((1-x-x^2)/(1-x-2x^2), x(1-x)/(1-2x)).

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 3, 2, 1, 3, 6, 6, 3, 1, 5, 14, 15, 10, 4, 1, 11, 30, 38, 29, 15, 5, 1, 21, 66, 92, 81, 49, 21, 6, 1, 43, 142, 220, 216, 150, 76, 28, 7, 1, 85, 306, 516, 560, 435, 253, 111, 36, 8, 1
Offset: 0

Views

Author

Paul Barry, Nov 11 2008

Keywords

Comments

Row sums are A061646. A147750=A007318^-1*A147747.

Examples

			Triangle begins
1,
0, 1,
1, 1, 1,
1, 3, 2, 1,
3, 6, 6, 3, 1,
5, 14, 15, 10, 4, 1,
11, 30, 38, 29, 15, 5, 1,
21, 66, 92, 81, 49, 21, 6, 1
Showing 1-3 of 3 results.

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