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

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

Author

Paul Barry, Nov 11 2008

Keywords

• easy
• nonn

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

• S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
• L. Edson Jeffery, Unit-primitive matrices.
• Index entries for linear recurrences with constant coefficients, signature (5,-5).

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