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

0,3

Links

Table of n, a(n) for n=0..54.

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.

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 0<k<=n; T(n, 0)=1; T(n, k)=0 for k>n.

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]

Example

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

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 *)

Crossrefs

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

Sequence in context: A279927 A137634 A100229 * A297506 A297720 A297654

Adjacent sequences: A071946 A071947 A071948 * A071950 A071951 A071952

Keyword

nonn,easy,tabl

Author

N. J. A. Sloane, Jun 15 2002

Extensions

Edited by Emeric Deutsch, Mar 04 2004

Status

approved