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A128899
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Riordan array (1,(1-2x-sqrt(1-4x))/(2x)).
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8
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1, 0, 1, 0, 2, 1, 0, 5, 4, 1, 0, 14, 14, 6, 1, 0, 42, 48, 27, 8, 1, 0, 132, 165, 110, 44, 10, 1, 0, 429, 572, 429, 208, 65, 12, 1, 0, 1430, 2002, 1638, 910, 350, 90, 14, 1, 0, 4862, 7072, 6188, 3808, 1700, 544, 119, 16, 1, 0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1
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OFFSET
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0,5
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COMMENTS
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Let the sequence A(n) = [0/1, 2/1, 1/2, 3/2, 2/3, 4/3, ...] defined by a(2n)=n/(n+1) and a(2n+1)=(n+2)/(n+1). T(n,k) is the triangle read by rows given by A(n) DELTA A000007 where DELTA is the operator defined in A084938.
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LINKS
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FORMULA
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T(n,k) = A039598(n-1,k-1) for n >= 1, k >= 1; T(n,0)=0^n.
T(n,k) = T(n-1,k-1) + 2*T(n-1,k) + T(n-1,k+1) for k >= 1, T(n,0)=0^n, T(n,k)=0 if k > n.
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EXAMPLE
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Triangle begins:
1;
0, 1;
0, 2, 1;
0, 5, 4, 1;
0, 14, 14, 6, 1;
0, 42, 48, 27, 8, 1;
0, 132, 165, 110, 44, 10, 1;
0, 429, 572, 429, 208, 65, 12, 1;
0, 1430, 2002, 1638, 910, 350, 90, 14, 1;
0, 4862, 7072, 6188, 3808, 1700, 544, 119, 16, 1;
0, 16796, 25194, 23256, 15504, 7752, 2907, 798, 152, 18, 1;
...
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MAPLE
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# Uses function PMatrix from A357368.
PMatrix(10, n -> binomial(2*n, n)/(n+1)); # Peter Luschny, Oct 19 2022
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MATHEMATICA
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T[n_, n_] := 1; T[_, 0] = 0; T[n_, k_] /; 0 < k < n := T[n, k] = T[n - 1, k - 1] + 2 T[n - 1, k] + T[n - 1, k + 1]; T[_, _] = 0;
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PROG
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(Sage)
@cached_function
def T(k, n):
if k==n: return 1
if k==0: return 0
return sum(catalan_number(i)*T(k-1, n-i) for i in (1..n-k+1))
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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