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A038730
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Path-counting triangular array T(i,j), read by rows, obtained from array t in A038792 by T(i,j) = t(2*i-j, j) (for i >= 1 and 1 <= j <= i).
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11
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1, 1, 2, 1, 4, 5, 1, 6, 12, 13, 1, 8, 23, 33, 34, 1, 10, 38, 73, 88, 89, 1, 12, 57, 141, 211, 232, 233, 1, 14, 80, 245, 455, 581, 609, 610, 1, 16, 107, 393, 888, 1350, 1560, 1596, 1597, 1, 18, 138, 593, 1594, 2881, 3805, 4135, 4180, 4181
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OFFSET
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1,3
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LINKS
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FORMULA
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T(n, n) = A001519(n) for n >= 1 (odd-indexed Fibonacci numbers).
Following Dil and Mezo (2008, p. 944), define the incomplete Fibonacci numbers by F(n,k) = Sum_{s = 0..k} binomial(n-1-s, s) for n >= 1 and 0 <= k <= floor((n-1)/2). Then T(i, j) = F(2*i-1, j-1) for 1 <= j <= i.
G.f. for column j: Define g(t,j) = ((1+t)^j * (1+t-t^2) + (1-t)^j * (1-t-t^2))/2, which is a function of t^2. Then the g.f. for column j is Sum_{i >= j} T(i,j)*x^i = x^j * (Fibonacci(2*j-1) * (1-x)^(j+1) + Fibonacci(2*j-2) * x * (1-x)^j - x * g(sqrt(x), j)) / ((1-x)^j * (1-3*x+x^2)). This follows from the results in Pintér and Srivastava (1999).
(End)
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EXAMPLE
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Triangle T(i,j) begins as follows:
1;
1, 2;
1, 4, 5;
1, 6, 12, 13;
1, 8, 23, 33, 34;
1, 10, 38, 73, 88, 89;
1, 12, 57, 141, 211, 232, 233;
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MAPLE
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t:= proc(i, j) option remember; `if`(i=1 or j=1, 1,
max(t(i-1, j)+t(i-1, j-1), t(i-1, j-1)+t(i, j-1)))
end:
T:= (i, j)-> t(2*i-j, j):
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MATHEMATICA
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T[i_, j_]:= Sum[Binomial[2i-k-2, k], {k, 0, j-1}];
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PROG
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(Magma) [(&+[Binomial(2*n-j-2, j): j in [0..k-1]]): k in [1..n], n in [1..12]]; // G. C. Greubel, Apr 05 2022
(SageMath)
def A038730(n, k): return sum( binomial(2*n-j-2, j) for j in (0..k-1))
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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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