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A177352
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The triangle t(n,k) of the binomial sum as in A177351 in the column index range -floor(n/2)-1 <=k <= floor(n/2)-1.
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0
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1, 1, 2, 2, 1, 3, 3, 2, 5, 5, 5, 4, 1, 8, 8, 8, 7, 3, 13, 13, 13, 13, 12, 7, 1, 21, 21, 21, 21, 20, 14, 4, 34, 34, 34, 34, 34, 33, 26, 11, 1, 55, 55, 55, 55, 55, 54, 46, 25, 5, 89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1
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OFFSET
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0,3
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COMMENTS
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Row sums are 1, 1, 5, 8, 20, 34, 72, 122, 241, 405, 769, 1284, 2375, 3947, 7165,
11866, 21238, 35078, 62094, 102340, 179561,.... which apparently is (n+1)*Fibonacci(n+1)- A129722(n) for even n, and n*Fibonacci(n+1)-A129722(n) for odd n.
The first column is A000045 by construction. The change in the column index range adds the Fibonacci numbers as a first column and removes the trailing zero in the rows compared to A177351.
Comment R. J. Mathar, Dec 20, 2010 (Start):
If we construct the complements of each row's entries with respect to the Fibonacci number of that row, an array
1; # complement to 2
1,4; # complement to 4,1
1,5 # complement to 7,3
1,6,12 # complement to 12,7,1
1,7,17 # complement to 20,14,4
1,8,23,33 # complement to 33,26,11,1
emerges which appears to be related to A038791. (End).
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LINKS
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Table of n, a(n) for n=0..60.
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EXAMPLE
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1
1;
2, 2, 1;
3, 3, 2;
5, 5, 5, 4, 1;
8, 8, 8, 7, 3;
13, 13, 13, 13, 12, 7, 1;
21, 21, 21, 21, 20, 14, 4;
34, 34, 34, 34, 34, 33, 26, 11, 1;
55, 55, 55, 55, 55, 54, 46, 25, 5;
89, 89, 89, 89, 89, 89, 88, 79, 51, 16, 1;
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MATHEMATICA
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w[n_, m_, k_] = Binomial[n - (m + k), m + k];
t[n_, k_] := Sum[w[n, m, k], {m, 1, Floor[n/2 - k]}];
Table[Table[t[n, k], {k, -Floor[n/2 + 1], Floor[n/2 + 1] - 2}], {n, 0,
10}]
Flatten[%]
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CROSSREFS
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Cf. A177351, A000045
Sequence in context: A011373 A321783 A327035 * A210798 A117501 A117915
Adjacent sequences: A177349 A177350 A177351 * A177353 A177354 A177355
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KEYWORD
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nonn,tabf
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AUTHOR
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Roger L. Bagula, Dec 10 2010
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STATUS
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approved
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