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Expansion of (1-x)/(1-4*x+3*x^2-x^3).
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%I #32 Aug 03 2016 11:30:16

%S 1,3,9,28,88,277,872,2745,8641,27201,85626,269542,848491,2670964,

%T 8407925,26467299,83316385,262271568,825604416,2598919345,8181135700,

%U 25753389181,81069068969,255197244033,803335158406,2528817970494

%N Expansion of (1-x)/(1-4*x+3*x^2-x^3).

%C Row sums of A124819.

%C Let M = a triangle with the triangular series in every column, but the leftmost column is shifted upwards one row. Then A124820 = Lim_{n->inf} M^n, the left-shifted vector considered as a sequence. - _Gary W. Adamson_, Jul 27 2010

%C Second trisection of Narayana's cows sequence A000930. - _Oboifeng Dira_, Aug 03 2016

%H Vincenzo Librandi, <a href="/A124820/b124820.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,1).

%F a(n) = sum( k=0..n, C(n+2k+1, 3k+1) ).

%F a(n) = A052529(n+1) - A052529(n), n>1. - _R. J. Mathar_, Dec 15 2008

%t CoefficientList[Series[(1 - x)/(1 - 4 x + 3 x^2 - x^3), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jun 20 2014 *)

%t LinearRecurrence[{4,-3,1},{1,3,9},30] (* _Harvey P. Dale_, Apr 29 2016 *)

%t Table[Sum[Binomial[n + 2 k + 1, 3 k + 1], {k, 0, n}], {n, 0, 25}] (* _Michael De Vlieger_, Aug 03 2016 *)

%o (PARI) a(n)=([0,1,0; 0,0,1; 1,-3,4]^n*[1;3;9])[1,1] \\ _Charles R Greathouse IV_, Aug 03 2016

%K nonn,easy

%O 0,2

%A _Paul Barry_, Nov 08 2006