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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. 1
 4, 20, 16, 176, 160, 64, 1876, 1808, 960, 256, 22064, 22048, 13248, 5120, 1024, 275568, 282528, 182528, 83456, 25600, 4096, 3584064, 3747456, 2542464, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. Row sums are A002894. Column 1 is A054474 LINKS Table of n, a(n) for n=1..55. Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, Cambridge Univ. Press, 2009, page 90. FORMULA G.f.: 1/( 1 - y*(1 - 1/A(x)) ) where A(x) is the o.g.f. for A002894. EXAMPLE 4, 20, 16, 176, 160, 64, 1876, 1808, 960, 256, 22064, 22048, 13248, 5120, 1024, 275568, 282528, 182528, 83456, 25600, 4096 MATHEMATICA 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 CROSSREFS Sequence in context: A213822 A182456 A196380 * A130316 A131745 A261755 Adjacent sequences: A227994 A227995 A227996 * A227998 A227999 A228000 KEYWORD nonn,walk,tabl AUTHOR Geoffrey Critzer, Oct 04 2013 STATUS approved

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Last modified September 7 23:15 EDT 2024. Contains 375749 sequences. (Running on oeis4.)