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A071949 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.

%I A071949 #16 Nov 15 2019 21:33:32
%S A071949 1,1,2,1,4,10,1,6,24,66,1,8,42,172,498,1,10,64,326,1360,4066,1,12,90,
%T A071949 536,2706,11444,34970,1,14,120,810,4672,23526,100520,312066,1,16,154,
%U A071949 1156,7410,42024,211546,911068,2862562,1,18,192,1582,11088,69002,387456,1951494,8457504,26824386
%N A071949 Triangle read by rows of numbers of paths in a lattice satisfying certain conditions.
%H A071949 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 A071949 T(n, k) = (n-k+1)*(Sum_{j=0..k-1} (2^(j+1)*binomial(k, j+1)*binomial(n+k, j)))/k for 0<k<=n; T(n, 0)=1; T(n, k)=0 for k>n.
%F A071949 T(n,0) = 1, T(n,n) = T(n,n-1) + T(n+1,n-1), otherwise T(n,k) = T(n,k-1) + T(n+1,k-1) + T(n-1,k). [_Gerald McGarvey_, Oct 09 2008]
%e A071949 Triangle begins:
%e A071949   1;
%e A071949   1, 2;
%e A071949   1, 4, 10;
%e A071949   1, 6, 24,  66;
%e A071949   1, 8, 42, 172, 498;
%e A071949   ...
%p A071949 T := proc(n,k) if k>0 and k<=n then (n-k+1)*sum(2^(j+1)*binomial(k,j+1)*binomial(n+k,j),j=0..k-1)/k elif k=0 then 1 else 0 fi end: seq(seq(T(n,k),k=0..n),n=0..10);
%t A071949 T[_, 0] = 1;
%t A071949 T[n_, n_] := T[n, n] = T[n, n-1] + T[n+1, n-1];
%t A071949 T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] + T[n+1, k-1] + T[n-1, k];
%t A071949 T[_, _] = 0;
%t A071949 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 15 2019 *)
%Y A071949 T(n, n)=A027307(n).
%K A071949 nonn,easy,tabl
%O A071949 0,3
%A A071949 _N. J. A. Sloane_, Jun 15 2002
%E A071949 Edited by _Emeric Deutsch_, Mar 04 2004