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%I #6 Sep 26 2021 01:54:09
%S 1,0,1,3,0,1,0,8,0,1,13,0,15,0,1,0,57,0,24,0,1,63,0,156,0,35,0,1,0,
%T 384,0,340,0,48,0,1,321,0,1380,0,645,0,63,0,1,0,2505,0,3800,0,1113,0,
%U 80,0,1,1683,0,11145,0,8855,0,1792,0,99,0,1,0,16008,0,37065,0,18368,0,2736,0
%N Triangle read by rows: T(n,k) is the number of paths in the right half-plane from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k h=(1,0) steps (0<=k<=n).
%C T(2n,0)=A001850(n) (the central Delannoy numbers); T(2n+1,0)=0. T(2n,1)=0; T(2n-1,1)=A108666(n). T(n,k)=0 if n+k is odd. Row sums yield A059345. See A132277 for the same statistic on paths restricted to the first quadrant.
%F G.f. = 1/sqrt((1-tz-z^2)^2-4z^2).
%e Triangle starts:
%e 1;
%e 0, 1;
%e 3, 0, 1;
%e 0, 8, 0, 1;
%e 13, 0, 15, 0, 1;
%e 0, 57, 0, 24, 0, 1;
%e T(3,1)=8 because we have hH, Hh, hUD, UhD, UDh, hDU, DhU and DUh.
%p G:=1/sqrt((1-t*z-z^2)^2-4*z^2): Gser:=simplify(series(G,z=0,15)): for n from 0 to 11 do P[n]:=sort(coeff(Gser,z,n)) end do: for n from 0 to 11 do seq(coeff(P[n],t,j),j=0..n) end do; # yields sequence in triangular form
%Y Cf. A001850, A108666, A059345, A132277.
%K nonn,tabl
%O 0,4
%A _Emeric Deutsch_, Sep 03 2007
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