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 A177352 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. 0
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS EXAMPLE 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; MATHEMATICA 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[%] CROSSREFS Cf. A177351, A000045 Sequence in context: A011373 A321783 A327035 * A210798 A117501 A117915 Adjacent sequences:  A177349 A177350 A177351 * A177353 A177354 A177355 KEYWORD nonn,tabf AUTHOR Roger L. Bagula, Dec 10 2010 STATUS approved

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Last modified April 19 00:18 EDT 2021. Contains 343098 sequences. (Running on oeis4.)