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A236076
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A skewed version of triangular array A122075.
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1
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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
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internal format)
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OFFSET
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0,3
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COMMENTS
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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.
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LINKS
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FORMULA
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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).
T(n,n) = Fibonacci(n+2) = A000045(n+2).
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EXAMPLE
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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;
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MATHEMATICA
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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 *)
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PROG
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(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
(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
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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