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A236076 A skewed version of triangular array A122075. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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.

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.

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).

EXAMPLE

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;

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 *)

PROG

(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

(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

CROSSREFS

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

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

Cf. A111006.

Sequence in context: A202502 A219839 A154312 * A119900 A328376 A141097

Adjacent sequences:  A236073 A236074 A236075 * A236077 A236078 A236079

KEYWORD

easy,nonn,tabl

AUTHOR

Philippe Deléham, Jan 19 2014

STATUS

approved

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Last modified October 22 14:44 EDT 2019. Contains 328318 sequences. (Running on oeis4.)