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

0,8

Comments

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

Antidiagonal sums: see A052946.

Links

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

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

Example

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;

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A111460 A035327 A004444 * A357079 A259790 A246654

Adjacent sequences: A204530 A204531 A204532 * A204534 A204535 A204536

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 16 2012

Status

approved