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

1, 1, 2, 1, 4, 10, 1, 6, 24, 66, 1, 8, 42, 172, 498, 1, 10, 64, 326, 1360, 4066, 1, 12, 90, 536, 2706, 11444, 34970, 1, 14, 120, 810, 4672, 23526, 100520, 312066, 1, 16, 154, 1156, 7410, 42024, 211546, 911068, 2862562, 1, 18, 192, 1582, 11088, 69002, 387456, 1951494, 8457504, 26824386

Offset: 0

Author

N. J. A. Sloane, Jun 15 2002

Keywords

• nonn
• easy
• tabl

Examples

			Triangle begins:
  1;
  1, 2;
  1, 4, 10;
  1, 6, 24,  66;
  1, 8, 42, 172, 498;
  ...
		

Links

• D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.

Crossrefs

T(n, n)=A027307(n).

Programs

Maple

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);

Mathematica

T[_, 0] = 1;
T[n_, n_] := T[n, n] = T[n, n-1] + T[n+1, n-1];
T[n_, k_] /; 0 <= k < n := T[n, k] = T[n, k-1] + T[n+1, k-1] + T[n-1, k];
T[, ] = 0;
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 15 2019 *)

Formula

T(n, k) = (n-k+1)*(Sum_{j=0..k-1} (2^(j+1)*binomial(k, j+1)*binomial(n+k, j)))/k for 0n.

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]

Extensions

Edited by Emeric Deutsch, Mar 04 2004