A016095 Triangular array T(n,k) read by rows, where T(n,k) = coefficient of x^n*y^k in 1/(1-x-y-(x+y)^2).

1, 1, 1, 2, 4, 2, 3, 9, 9, 3, 5, 20, 30, 20, 5, 8, 40, 80, 80, 40, 8, 13, 78, 195, 260, 195, 78, 13, 21, 147, 441, 735, 735, 441, 147, 21, 34, 272, 952, 1904, 2380, 1904, 952, 272, 34, 55, 495, 1980, 4620, 6930, 6930, 4620, 1980, 495, 55

Offset

0,4

Comments

Triangle T(n,k), 0<=k<=n, read by rows, given by [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 10 2005

Links

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

Formula

G.f.: 1/(1-x-y-(x+y)^2).

T(n,k) = Fibonacci(n+1)*binomial(n,k) = A000045(n+1)*A007318(n,k). - Philippe Deléham, Oct 14 2006

Sum_{k=0..floor(n/2)} T(n-k,k) = A123392(n). - Philippe Deléham, Oct 14 2006

G.f.: T(0)/2, where T(k) = 1 + 1/(1 - (2*k+1+ x*(1+y))*x*(1+y)/((2*k+2+ x*(1+y))*x*(1+y)+ 1/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013

T(n,k) = T(n-1,k)+T(n-1,k-1)+T(n-2,k)+2*T(n-2,k-1)+T(n-2,k-2), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = T(2,2) = 2, T(2,1) = 4, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 12 2013

Example

Triangle begins:
  1;
  1,  1;
  2,  4,  2;
  3,  9,  9,  3;
  5, 20, 30, 20,  5;
  8, 40, 80, 80, 40, 8;
  ...

Maple

read transforms; 1/(1-x-y-(x+y)^2); SERIES2(%, x, y, 12); SERIES2TOLIST(%, x, y, 12);

Mathematica

T[n_, k_] := SeriesCoefficient[1/(1-x-y-(x+y)^2), {x, 0, n}, {y, 0, k}]; Table[T[n-k, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 04 2017 *)

Crossrefs

Columns include A000045, A023607. Central diagonal is A102307. Antidiagonal sums are in A063727.

Sequence in context: A335678 A368434 A134400 * A298309 A349205 A181399

Adjacent sequences: A016092 A016093 A016094 * A016096 A016097 A016098

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Jan 23 2001

Status

approved