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A071945 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).

%I A071945 #21 Apr 25 2020 01:25:02
%S A071945 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 A071945 431,431,1,13,73,249,609,1153,1697,1697,1,15,99,399,1151,2591,4719,
%U A071945 6847,6847,1,17,129,601,2001,5201,11073,19617,28161,28161,1,19,163,863,3263
%N A071945 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 A071945 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 A071945 Peter Kagey, <a href="/A071945/b071945.txt">Table of n, a(n) for n = 0..8255</a> (first 128 rows, flattened)
%H A071945 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 A071945 G.f.: (1-q)/[z(1+tz)(2t-1+q)], where q=sqrt(1-4tz-4t^2z^2).
%e A071945 a(3,1)=5 because we have RRRV, RRVR, RVRR, RD and DR.
%e A071945 Triangle begins:
%e A071945 1
%e A071945 1 1
%e A071945 1 3    3
%e A071945 1 5    9   9
%e A071945 1 7   19  31   31
%e A071945 1 9   33  73  113  113
%e A071945 1 11  51 143  287  431   431
%e A071945 1 13  73 249  609 1153  1697  1697
%e A071945 1 15  99 399 1151 2591  4719  6847  6847
%e A071945 1 17 129 601 2001 5201 11073 19617 28161 28161
%Y A071945 Diagonal entries give A052709.
%K A071945 nonn,easy,tabl
%O A071945 0,5
%A A071945 _N. J. A. Sloane_, Jun 15 2002
%E A071945 Edited by _Emeric Deutsch_, Dec 21 2003