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A147748
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Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).
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7
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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
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OFFSET
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0,2
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COMMENTS
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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)
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LINKS
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Table of n, a(n) for n=0..24.
S. Felsner, D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
L. E. Jeffery, Unit-primitive matrices
Index entries for linear recurrences with constant coefficients, signature (5,-5).
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FORMULA
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G.f.: (1-3x+x^2)/(1-5x+5x^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) = 0.4*((5-sqrt5)/2)^n +0.4*((5+sqrt5)/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
a(n) = A030191(n)-3*A030191(n-1)+A030191(n-2). a(n) = 2*A081567(n-1), n>0. - R. J. Mathar, Apr 20 2011
a(n) = Sum_{k=0..n} A147746(n,k)*2^k. - Philippe Deléham, Oct 30 2011
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MAPLE
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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
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MATHEMATICA
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(1 - 3x + x^2)/(1 - 5x + 5x^2) + O[x]^25 // CoefficientList[#, x]& (* Jean-François Alcover, Oct 05 2016 *)
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CROSSREFS
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Cf. A033191, A081567, A178381, A189315.
Sequence in context: A229472 A135413 A193653 * A150125 A224514 A065345
Adjacent sequences: A147745 A147746 A147747 * A147749 A147750 A147751
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KEYWORD
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easy,nonn
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AUTHOR
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Paul Barry, Nov 11 2008
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STATUS
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approved
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