A102756 Triangle T(n,k), 0<=k<=n, read by rows defined by: T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-2,k-2) - T(n-2,k), T(0,0) = 1, T(n,k) = 0 if k < 0 or if n < k.

1, 2, 1, 3, 4, 2, 4, 10, 10, 3, 5, 20, 31, 20, 5, 6, 35, 76, 78, 40, 8, 7, 56, 161, 232, 184, 76, 13, 8, 84, 308, 582, 636, 406, 142, 21, 9, 120, 546, 1296, 1831, 1604, 861, 260, 34, 10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55

Offset: 0

Author

Philippe Deléham, Dec 18 2006

Keywords

• nonn
• tabl

Comments

Rising and falling diagonals are A008999, A124400.

Subtriangle of triangle given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 17 2012

Jointly generated with A209130 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 05 2012

Examples

			Triangle begins:
  1;
  2, 1;
  3, 4, 2;
  4, 10, 10, 3;
  5, 20, 31, 20, 5;
  6, 35, 76, 78, 40, 8;
  7, 56, 161, 232, 184, 76, 13;
  8, 84, 308, 582, 636, 406, 142, 21;
  9, 120, 546, 1296, 1831, 1604, 861, 260, 34;
  10, 165, 912, 2640, 4630, 5215, 3820, 1766, 470, 55;
Triangle (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins:
  1
  1, 0
  2, 1, 0
  3, 4, 2, 0
  4, 10, 10, 3, 0
  5, 20, 31, 20, 5, 0
  6, 35, 76, 78, 40, 8, 0
  7, 56, 161, 232, 184, 76, 13, 0
		

Crossrefs

Cf. A209130.

Cf. A008999, A124400, A084938, A209130.

Cf. A254006, A000012, A000027, A000244, A015530, A015544.

Cf. A001629, A000045, A000292, A051747.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A102756 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A209130 *)
(* Clark Kimberling, Mar 05 2012 *)

Formula

Sum_{k=0..n} x^k*T(n,k) = A254006(n), A000012(n), A000027(n+1), A000244(n), A015530(n+1), A015544(n+1) for x = -2, -1, 0, 1, 2, 3 respectively.

T(n,n-1) = 2*A001629(n+1) for n>=1.

T(n,n) = Fibonacci(n+1) = A000045(n+1).

T(n,0) = n+1.

T(n,1) = A000292(n) for n>=1.

T(n+1,2) = binomial(n+4,n-1)+binomial(n+2,n-1)= A051747(n) for n>=1.

G.f.: 1/(1-(2+y)*x+(1+y)*(1-y)*x^2). - Philippe Deléham, Feb 17 2012