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%I #21 Aug 10 2015 23:52:29
%S 0,0,1,5,19,61,180,502,1349,3529,9050,22854,57014,140832,345036,
%T 839530,2030757,4887423,11710757,27951471,66486128,157661282,
%U 372840407,879510801,2070045268,4862121660,11398688956,26676792832,62333380456,145434747140
%N Expansion of (x^2)/((1-x)*(1-2*x-x^2+x^3)^2)
%C Second of a series of sequences of partial sums of (nonzero) diagonals of triangle A188106 whose diagonals correspond to successive convolutions of A006054 with itself, where the first such sequence of partial sums is given by A077850. For n=1,2,..., this series of sequences is generated by successive series expansion of 1/((1-x)*(1-2*x-x^2+x^3)^n), for which A077850 corresponds to n=1 and A189427 corresponds to n=2.
%C a(n)=Sum_{k=0..n} A189426(k), where A189426={0,0,1,4,14,42,119,322,...} is the convolution of A006054={0,0,1,2,5,11,25,56,126,...} with itself. Also, a(n+2)=Sum_{k=0..n} A188106{n+k+1,k}, n=0,1,2,....
%F G.f.: (x^2)/((1-x)*(1-2*x-x^2+x^3)^2).
%F a(n)=5*a(n-1)-6*a(n-2)-4*a(n-3)+9*a(n-4)-a(n-5)-3*a(n-6)+a(n-7), n>=7, a{m}={0,0,1,5,19,61,180}, m=0..6.
%o (PARI) Vec((x^2)/((1-x)*(1-2*x-x^2+x^3)^2)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 26 2012
%Y Cf. A006054, A077850, A188106, A189426.
%K nonn,easy
%O 0,4
%A _L. Edson Jeffery_, Apr 22 2011
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