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 A189427 #23 Jun 02 2025 04:02:16
%S A189427 0,0,1,5,19,61,180,502,1349,3529,9050,22854,57014,140832,345036,
%T A189427 839530,2030757,4887423,11710757,27951471,66486128,157661282,
%U A189427 372840407,879510801,2070045268,4862121660,11398688956,26676792832,62333380456,145434747140
%N A189427 Expansion of (x^2)/((1-x)*(1-2*x-x^2+x^3)^2).
%C A189427 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 A189427 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 A189427 G.f.: (x^2)/((1-x)*(1-2*x-x^2+x^3)^2).
%F A189427 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 A189427 (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 A189427 Cf. A006054, A077850, A188106, A189426.
%K A189427 nonn,easy
%O A189427 0,4
%A A189427 _L. Edson Jeffery_, Apr 22 2011