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A227997 Triangular array read by rows. T(n,k) is the number of square lattice walks that start and end at the origin after 2n steps having k primitive loops; n>=1, 1<=k<=n.

%I A227997 #32 Apr 26 2016 12:32:12
%S A227997 4,20,16,176,160,64,1876,1808,960,256,22064,22048,13248,5120,1024,
%T A227997 275568,282528,182528,83456,25600,4096,3584064,3747456,2542464,
%U A227997 1284096,481280,122880,16384,47995476,50981136,35851968,19365120,8186880,2617344,573440,65536,657037232,707110432,511288256,290053120,133084160,48799744,13647872,2621440,262144,9150655216,9958458656,7363711104,4338317824,2113592320,851398656,276856832,68943872,11796480,1048576
%N A227997 Triangular array read by rows.  T(n,k) is the number of square lattice walks that start and end at the origin after 2n steps having k primitive loops; n>=1, 1<=k<=n.
%C A227997 The walk consists of steps in the four directions NW,NE,SW,SE. A primitive loop is a walk that starts and ends at the origin but does not otherwise touch the origin.
%C A227997 Row sums are A002894.
%C A227997 Column 1 is A054474
%H A227997 Philippe Flajolet and Robert Sedgewick, <a href="http://algo.inria.fr/flajolet/Publications/AnaCombi/anacombi.html">Analytic Combinatorics</a>, Cambridge Univ. Press, 2009, page 90.
%F A227997 G.f.: 1/( 1 - y*(1 - 1/A(x)) ) where A(x) is the o.g.f. for A002894.
%e A227997 4,
%e A227997 20, 16,
%e A227997 176, 160, 64,
%e A227997 1876, 1808, 960, 256,
%e A227997 22064, 22048, 13248, 5120, 1024,
%e A227997 275568, 282528, 182528, 83456, 25600, 4096
%t A227997 nn=6;a=Sum[Binomial[2n,n]^2x^n,{n,0,nn}];Map[Select[#,#>0&]&,Drop[CoefficientList[Series[1/(1-y(1-1/a)),{x,0,nn}],{x,y}],1]]//Grid
%K A227997 nonn,walk,tabl
%O A227997 1,1
%A A227997 _Geoffrey Critzer_, Oct 04 2013