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%I #21 Apr 25 2020 01:25:02
%S 1,1,1,1,3,3,1,5,9,9,1,7,19,31,31,1,9,33,73,113,113,1,11,51,143,287,
%T 431,431,1,13,73,249,609,1153,1697,1697,1,15,99,399,1151,2591,4719,
%U 6847,6847,1,17,129,601,2001,5201,11073,19617,28161,28161,1,19,163,863,3263
%N Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using steps R=(1,0), V=(0,1) and D=(2,1).
%C Also could be titled: "Table read by antidiagonals upward: T(n,k) is the number of ways to move a chess king from (1,1) to (n,k) in the first quadrant using only right, diagonal up-right, and diagonal up-left moves." - _Peter Kagey_, Apr 20 2020
%H Peter Kagey, <a href="/A071945/b071945.txt">Table of n, a(n) for n = 0..8255</a> (first 128 rows, flattened)
%H D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.4153/CJM-1997-015-x">On some alternative characterizations of Riordan arrays</a>, Canad J. Math., 49 (1997), 301-320.
%F G.f.: (1-q)/[z(1+tz)(2t-1+q)], where q=sqrt(1-4tz-4t^2z^2).
%e a(3,1)=5 because we have RRRV, RRVR, RVRR, RD and DR.
%e Triangle begins:
%e 1
%e 1 1
%e 1 3 3
%e 1 5 9 9
%e 1 7 19 31 31
%e 1 9 33 73 113 113
%e 1 11 51 143 287 431 431
%e 1 13 73 249 609 1153 1697 1697
%e 1 15 99 399 1151 2591 4719 6847 6847
%e 1 17 129 601 2001 5201 11073 19617 28161 28161
%Y Diagonal entries give A052709.
%K nonn,easy,tabl
%O 0,5
%A _N. J. A. Sloane_, Jun 15 2002
%E Edited by _Emeric Deutsch_, Dec 21 2003
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