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.
%I A147748 #53 Jul 10 2024 05:29:51
%S A147748 1,2,6,20,70,250,900,3250,11750,42500,153750,556250,2012500,7281250,
%T A147748 26343750,95312500,344843750,1247656250,4514062500,16332031250,
%U A147748 59089843750,213789062500,773496093750,2798535156250,10125195312500
%N A147748 Row sums of Riordan array ((1-3x+x^2)/(1-4x+3x^2), x(1-2x)/(1-4x+3x^2)).
%C A147748 Row sums of A147747. Binomial transform of A061646.
%C A147748 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
%C A147748 From _L. Edson Jeffery_, Apr 19 2011: (Start)
%C A147748 For the 5 X 5 unit-primitive matrix (see [Jeffery])
%C A147748 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],
%C A147748 a(n) = (Trace([A_(10,1)]^(2*n)))/5. (See also A189315.) (End)
%H A147748 S. Felsner and D. Heldt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Felsner/felsner2.html">Lattice Path Enumeration and Toeplitz Matrices</a>, J. Int. Seq. 18 (2015) # 15.1.3.
%H A147748 L. Edson Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.
%H A147748 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%F A147748 G.f.: (1-3*x+x^2)/(1-5*x+5*x^2).
%F A147748 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
%F A147748 For n >= 1: a(n) = (2/5)*((5-sqrt(5))/2)^n + (2/5)*((5+sqrt(5))/2)^n. - _Richard Choulet_, Nov 14 2008
%F A147748 G.f.: 1/(1-2x/(1-x/(1-x/(1-x)))) (hence sequence approximates A000984 in first few terms). - _Paul Barry_, Aug 05 2009
%F A147748 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
%F A147748 From _R. J. Mathar_, Apr 20 2011: (Start)
%F A147748 a(n) = A030191(n) - 3*A030191(n-1) + A030191(n-2).
%F A147748 a(n) = 2*A081567(n-1), n > 0. (End)
%F A147748 a(n) = Sum_{k=0..n} A147746(n,k)*2^k. - _Philippe Deléham_, Oct 30 2011
%F A147748 E.g.f.: (1 + 4*exp(5*x/2)*cosh(sqrt(5)*x/2))/5. - _Stefano Spezia_, Jul 09 2024
%p A147748 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
%t A147748 (1 - 3x + x^2)/(1 - 5x + 5x^2) + O[x]^25 // CoefficientList[#, x]& (* _Jean-François Alcover_, Oct 05 2016 *)
%Y A147748 Cf. A000984, A030191, A061646, A081567, A147746, A147747, A178381, A189315.
%K A147748 easy,nonn
%O A147748 0,2
%A A147748 _Paul Barry_, Nov 11 2008