A236076 A skewed version of triangular array A122075.

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 1, 7, 8, 0, 0, 0, 4, 15, 13, 0, 0, 0, 1, 12, 30, 21, 0, 0, 0, 0, 5, 31, 58, 34, 0, 0, 0, 0, 1, 18, 73, 109, 55, 0, 0, 0, 0, 0, 6, 54, 162, 201, 89, 0, 0, 0, 0, 0, 1, 25, 145, 344, 365, 144, 0, 0, 0, 0, 0, 0, 7, 85, 361

Offset: 0

Author

Philippe Deléham, Jan 19 2014

Keywords

• easy
• nonn
• tabl

Comments

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

Subtriangle of the triangle A122950.

Examples

			Triangle begins:
  1;
  0,  2;
  0,  1,  3;
  0,  0,  3,  5;
  0,  0,  1,  7,  8;
  0,  0,  0,  4, 15, 13;
  0,  0,  0,  1, 12, 30, 21;
  0,  0,  0,  0,  5, 31, 58, 34;
		

Links

• Reinhard Zumkeller, Rows n = 0..125 of triangle, flattened
• H. Fuks and J.M.G. Soto, Exponential convergence to equilibrium in cellular automata asymptotically emulating identity, arXiv:1306.1189 [nlin.CG], 2013.

Crossrefs

Cf. variant: A055830, A122075, A122950, A208337.

Cf. A167704 (diagonal sums), A000079 (row sums).

Cf. A111006.

Programs

Haskell

a236076 n k = a236076_tabl !! n !! k
a236076_row n = a236076_tabl !! n
a236076_tabl = [1] : [0, 2] : f [1] [0, 2] where
   f us vs = ws : f vs ws where
     ws = [0] ++ zipWith (+) (zipWith (+) ([0] ++ us) (us ++ [0])) vs
-- Reinhard Zumkeller, Jan 19 2014

Mathematica

T[n_, k_]:= If[k<0 || k>n, 0, If[n==0 && k==0, 1, If[k==0, 0, If[n==1 && k==1, 2, T[n-1, k-1] + T[n-2, k-1] + T[n-2, k-2]]]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 21 2019 *)

PARI

{T(n,k) = if(k<0 || k>n, 0, if(n==0 && k==0, 1, if(k==0, 0, if(n==1 && k==1, 2, T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2) ))))}; \\ G. C. Greubel, May 21 2019

Sage

def T(n, k):
    if (k<0 or k>n): return 0
    elif (n==0 and k==0): return 1
    elif (k==0): return 0
    elif (n==1 and k==1): return 2
    else: return T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2)
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 21 2019

Formula

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

T(n,k) = T(n-1,k-1) + T(n-2,k-1) + T(n-2,k-2), T(0,0)=1, T(1,0) = 0, T(1,1) = 2, T(n,k) = 0 if k < 0 or if k > n.

Sum_{k=0..n} T(n,k) = 2^n = A000079(n).

Sum_{n>=k} T(n,k) = A078057(k) = A001333(k+1).

T(n,n) = Fibonacci(n+2) = A000045(n+2).

T(n+1,n) = A023610(n-1), n >= 1.

T(n+2,n) = A129707(n).