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A199881
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Triangle T(n,k), read by rows, given by (1,-1,0,0,0,0,0,0,0,0,0,...) DELTA (1,0,-1,1,0,0,0,0,0,0,0,...) where DELTA is the operator defined in A084938.
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2
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1, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 3, 1, 0, 0, 1, 4, 4, 1, 0, 0, 0, 3, 7, 5, 1, 0, 0, 0, 1, 7, 11, 6, 1, 0, 0, 0, 0, 4, 14, 16, 7, 1, 0, 0, 0, 0, 1, 11, 25, 22, 8, 1, 0, 0, 0, 0, 0, 5, 25, 41, 29, 9, 1, 0, 0, 0, 0, 0, 1, 16, 50, 63, 37, 10, 1
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OFFSET
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0,9
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COMMENTS
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The nonzero entries of column k give row k+1 in A072405.
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LINKS
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FORMULA
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T(n,k) = T(n-1,k-1) + T(n-2,k-1) starting with T(0,0) = T(1,0) = T(1,1) = T(2,1) = T(2,2) = 1 and T(2,0) = 0.
G.f.: (1+x-y*x^2)/(1-y*x-y*x^2).
T(n, k) = binomial(k, n-k) + binomial(k+1, n-k-1).
T(n, k) = (-1)^(n-k)*A104402(n, k). (End)
Sum_{k=0..n} T(n, k) = 2*Fibonacci(n) + [n=0].
Sum_{n=k..2*k+1} T(n,k) = 3*2^(n-1) + (1/2)*[n=0]. (End)
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EXAMPLE
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Triangle begins:
1;
1, 1;
0, 1, 1;
0, 1, 2, 1; (key row for starting the recurrence)
0, 0, 2, 3, 1;
0, 0, 1, 4, 4, 1;
0, 0, 0, 3, 7, 5, 1;
0, 0, 0, 1, 7, 11, 6, 1;
0, 0, 0, 0, 4, 14, 16, 7, 1;
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MATHEMATICA
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T[n_, k_]:= T[n, k]= If[n<2, 1, If[k==0, 0, If[k==n, 1, If[n==2 && k==1, 1, T[n-1, k-1] +T[n-2, k-1] ]]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 28 2021 *)
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PROG
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(Sage)
def T(n, k): return binomial(k, n-k) + binomial(k+1, n-k-1)
flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 28 2021
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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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