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A327917
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Triangle T read by rows: T(k, n) = A(k-n, k) with the array A(k, n) = F(2*k+n) = A000045(2*k+n), for k >= 0 and n >= 0.
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0
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0, 1, 1, 3, 2, 1, 8, 5, 3, 2, 21, 13, 8, 5, 3, 55, 34, 21, 13, 8, 5, 144, 89, 55, 34, 21, 13, 8, 377, 233, 144, 89, 55, 34, 21, 13, 987, 610, 377, 233, 144, 89, 55, 34, 21, 2584, 1597, 987, 610, 377, 233, 144, 89, 55, 34, 6765, 4181, 2584, 1597, 987, 610, 377, 233, 144, 89, 55
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OFFSET
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0,4
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COMMENTS
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This is the row reversed triangle A199334.
This is the analog of the array and the triangle A327916, where the positive odd numbers instead of the Fibonacci numbers are used.
The array A arises from the following Pascal-type triangles PF(k), for k >= 0, based on A000045 (Fibonacci). For example, the Pascal type triangle PF(k), for k = 3 is
0 1 1 2
1 2 3
3 5
8
For k -> infinity the left-aligned row sequences build the array A(k, n), with k >= 0 and n >= 0, that is, A(k, n) = F(2*k + n). See the example section for the first rows of A.
The first column sequence of A, {F(2*n) = A001906(n)}_{n>=0}, is the binomial transform of the first (k=0) row sequence of A, {F(n)}_{n>=0}.
The triangle T is the array A read by upwards antidiagonals.
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LINKS
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FORMULA
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A(k, n) = Sum_{j=0..k} binomial(k, j)*F(n+j) = F(2*k+n), for k >= 0 and n >= 0.
T(k, n) = A(k - n, n) = F(2*k - n), for k >= 0 and n = 0..k, with the Fibonacci numbers F = A000045.
Recurrence: T(k,0) = F(2*k), k >= 0, T(k, n) = T(k, n-1) - T(k-1, n-1), k >= 1, n = 1..k, and T(k, n) = 0 if k < n.
O.g.f. for row polynomials R(n, x) = Sum_{n=0..k} T(k, n)*x^n:
G(x, z) = Sum_{n=0} R(n, x)*z^n = z*(1 + x - 2*x*z)/((1 - 3*z + z^2)*(1 - x*z - (x*z)^2)).
T(k, 0) = Sum_{n=0..k} binomial(k,n)*T(n, n), k >= 0 (binomial transform).
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EXAMPLE
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The Array A(k, n) begins:
k\n 0 1 2 3 4 5 ...
-----------------------------
0: 0 1 1 2 3 5 ... F(n)
1: 1 2 3 5 8 13 ... F(n+2)
2: 3 5 8 13 21 34 ... F(n+4)
3: 8 13 21 34 55 89 ... F(n+6)
4: 21 34 55 89 144 233 ... F(n+8)
5: 55 89 144 233 377 610 ... F(n+10)
...
---------------------------------------
The triangle T(k, n) begins:
k\n 0 1 2 3 4 5 6 7 8 9 10 ...
------------------------------------------------------
0: 0
1: 1 1
2: 3 2 1
3: 8 5 3 2
4: 21 13 8 5 3
5: 55 34 21 13 8 5
6: 144 89 55 34 21 13 8
7: 377 233 144 89 55 34 21 13
8: 987 610 377 233 144 89 55 34 21
9: 2584 1597 987 610 377 233 144 89 55 34
10: 6765 4181 2584 1597 987 610 377 233 144 89 55
...
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CROSSREFS
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Column sequences of T (no leading zeros) and A: from the shifted Fibonacci bisection {F(2*k) = A001906(k)} for even n, and {F(2*k+1) = A001519(k+1)}, for odd n.
Row sums: 2*A094292(n+1) = F(2*(n+1)) - F(n+1), n >= 0.
Alternating row sums: 2*A164267(n-1), n >= 0, with 0 for n = 0.
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KEYWORD
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AUTHOR
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STATUS
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approved
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