A204533 Triangle T(n,k), read by rows, given by (0, 1, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 8, 7, 3, 1, 0, 21, 22, 12, 4, 1, 0, 55, 67, 43, 18, 5, 1, 0, 144, 200, 147, 72, 25, 6, 1, 0, 377, 588, 486, 271, 110, 33, 7, 1, 0, 987, 1708, 1566, 976, 450, 158, 42, 8, 1

Offset: 0

Author

Philippe Deléham, Jan 16 2012

Keywords

• nonn
• tabl

Comments

Riordan array (1, x*(1-x)^2/(1-3*x+x^2)).

Antidiagonal sums: see A052946.

Examples

			Triangle begins:
  1;
  0,   1;
  0,   1,   1;
  0,   3,   2,   1;
  0,   8,   7,   3,  1;
  0,  21,  22,  12,  4,  1;
  0,  55,  67,  43, 18,  5, 1;
  0, 144, 200, 147, 72, 25, 6, 1;
		

Links

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

Crossrefs

Cf. diagonals: A000007, A088305, A000012, A001477, A055998.

Programs

Mathematica

Table[Sum[Binomial[k, m - 1] Binomial[n - 2 m + k, n - k - 1], {k, 0, n - 1}] + Boole[n == m == 0], {n, 0, 9}, {m, 0, n}] // Flatten (* Michael De Vlieger, Sep 26 2018 *)

Maxima

T(n,m):= if n=0 and m=0 then 1 else sum(binomial(k,m-1)*binomial(n-2*m+k,n-k-1),k,0,n-1); /* Vladimir Kruchinin, Sep 27 2018 */

PARI

T(n,k) = if ((n==0) && (k==0), 1, sum(i=0, n-1, binomial(i,k-1)*binomial(n-2*k+i,n-i-1))); \\ Michel Marcus, Sep 27 2018

Formula

Sum_{k=0..n} T(n,k) = A204200(n+1).

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

G.f.: (-1 + 3*x - x^2)/(-1 + 3*x - x^2 + x*y - 2*x^2*y + x^3*y). - R. J. Mathar, Aug 11 2015

T(n,m) = Sum_{k=0..n-1} C(k,m-1)*C(n-2*m+k,n-k-1), T(0,0)=1. - Vladimir Kruchinin, Sep 27 2018