A069270 Third level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 2nd level is A069269).

1, 1, 1, 1, 2, 4, 1, 3, 9, 22, 1, 4, 15, 52, 140, 1, 5, 22, 91, 340, 969, 1, 6, 30, 140, 612, 2394, 7084, 1, 7, 39, 200, 969, 4389, 17710, 53820, 1, 8, 49, 272, 1425, 7084, 32890, 135720, 420732, 1, 9, 60, 357, 1995, 10626, 53820, 254475, 1068012, 3362260

Offset

0,5

Comments

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).

Antidiagonals of convolution matrix of Table 1.5, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

Links

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)

V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.

Formula

T(n, k) = C(n+3k, k)*(n-k+1)/(n+2k+1).

For n >= k+3: T(n, k) = T(n-2, k+1)-T(n-3, k+1).

T(n, n) = T(n+2, n-1) = C(4n, n)/(3n+1).

Example

Rows start
  1;
  1,   1;
  1,   2,   4;
  1,   3,   9,  22;
  1,   4,  15,  52, 140;
etc.

Maple

A069270 := proc(n, k)
        binomial(n+3*k, k)*(n-k+1)/(n+2*k+1) ;
end proc: # R. J. Mathar, Oct 11 2015

Mathematica

Table[Binomial[n + 3 k, k] (n - k + 1)/(n + 2 k + 1), {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 27 2019 *)

Crossrefs

Columns include A000012, A000027, A055999.

Right-hand diagonals include A002293, A069271, A006632.

Cf. A130458 (row sums).

Sequence in context: A209573 A100075 A059836 * A079901 A121426 A190183

Adjacent sequences: A069267 A069268 A069269 * A069271 A069272 A069273

Keyword

nonn,tabl,easy,changed

Author

Henry Bottomley, Mar 12 2002

Status

approved