cp's OEIS Frontend

A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).

%I A071946 #28 Jan 25 2025 08:34:56
%S A071946 1,1,1,1,2,2,1,4,6,6,1,6,13,19,19,1,8,23,44,63,63,1,10,37,87,156,219,
%T A071946 219,1,12,55,155,330,568,787,787,1,14,77,255,629,1260,2110,2897,2897,
%U A071946 1,16,103,395,1111,2527,4856,7972,10869,10869,1,18,133,583,1849,4706,10130,18889,30545,41414,41414
%N A071946 Triangle T(n,k) read by rows giving number of underdiagonal lattice paths from (0,0) to (n,k) using only steps R = (1,0), V = (0,1) and D = (3,1).
%H A071946 Alois P. Heinz, <a href="/A071946/b071946.txt">Rows n = 0..150, flattened</a>
%H A071946 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.
%e A071946 Triangle T(n,k) begins:
%e A071946   1;
%e A071946   1, 1;
%e A071946   1, 2,  2;
%e A071946   1, 4,  6,  6;
%e A071946   1, 6, 13, 19, 19;
%e A071946   ...
%p A071946 T:= proc(n, k) option remember; `if`(n=0 and k=0, 1,
%p A071946      `if`(k<0 or n<k, 0, T(n-1, k)+T(n, k-1)+T(n-3, k-1)))
%p A071946     end:
%p A071946 seq(seq(T(n,k), k=0..n), n=0..12);  # _Alois P. Heinz_, May 05 2023
%t A071946 T[n_, k_] := T[n, k] = If[n == 0 && k == 0, 1,
%t A071946    If[k < 0 || n < k, 0, T[n-1, k] + T[n, k-1] + T[n-3, k-1]]];
%t A071946 Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Jan 25 2025, after _Alois P. Heinz_ *)
%Y A071946 Related arrays: A071943, A071944, A071945.
%Y A071946 A108076 is the reverse, A119254 is the row sums and A071969 is the last (largest) number in each row.
%K A071946 nonn,easy,tabl
%O A071946 0,5
%A A071946 _N. J. A. Sloane_, Jun 15 2002
%E A071946 More terms from _Joshua Zucker_, May 10 2006