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A140725 Inverse binomial transform of (0 followed by A037481).

%I A140725 #16 Jul 14 2025 10:10:35
%S A140725 0,1,4,10,34,94,298,862,2650,7822,23722,70654,212986,636910,1914826,
%T A140725 5736286,17225242,51642958,154994410,464852158,1394818618,4183931566,
%U A140725 12552843274,37656432670,112973492314,338912088334,1016753042218
%N A140725 Inverse binomial transform of (0 followed by A037481).
%C A140725 From _Sean A. Irvine_, Jun 07 2025: (Start)
%C A140725 For n>=1, the number of walks of length n-1 starting at vertex 1 (or, by symmetry, vertex 4) in the graph K_{1,1,3}:
%C A140725     1---2
%C A140725    /|\ /
%C A140725   0 | X
%C A140725    \|/ \
%C A140725     4---3. (End)
%H A140725 Sean A. Irvine, <a href="https://oeis.org/wiki/User:Sean_A._Irvine/Walks_on_Graphs#5_vertices">Walks on Graphs</a>.
%F A140725 a(n)= (-1)^n*A091003(n), n>0.
%F A140725 a(n+1)-3*a(n) = (-1)^(n+1)*A000079(n-1), n>0.
%F A140725 |a(n+1)-3*a(n)| = A011782(n).
%F A140725 From _R. J. Mathar_, Jul 14 2008: (Start)
%F A140725 O.g.f.: (1+3*x)*x / ((1+2*x)*(1-3*x)).
%F A140725 a(n) = ((-2)^n+4*3^n)/10, n>0. (End)
%F A140725 a(n) = a(n-1)+6*a(n-2) for n>2, a(0)=0, a(1)=1, a(2)=4. - _Philippe Deléham_, Nov 17 2013
%F A140725 a(n) + a(n+1) = A140796(n). - _Philippe Deléham_, Nov 17 2013
%F A140725 a(n+1) = sum_{k=0..n} A108561(n,k)*(-3)^k. - _Philippe Deléham_, Nov 17 2013
%t A140725 Join[{0},LinearRecurrence[{1,6},{1,4},26]] (* or *) a[0]=0;a[n_]:= ((-2)^n+4*3^n)/10;Array[a,27,0] (* _James C. McMahon_, Jul 13 2025 *)
%Y A140725 Cf. A083421 (bin. transform of (0 followed by A037481)).
%K A140725 nonn
%O A140725 0,3
%A A140725 _Paul Curtz_, Jul 12 2008
%E A140725 Edited and extended by _R. J. Mathar_, Jul 14 2008