A354267 A Fibonacci-Pascal triangle read by rows: T(n, n) = 1, T(n, n-1) = n - 1, T(n, 0) = T(n-1, 1) and T(n, k) = T(n-1, k-1) + T(n-1, k) for 0 < k < n-1.
1, 0, 1, 1, 1, 1, 1, 2, 2, 1, 2, 3, 4, 3, 1, 3, 5, 7, 7, 4, 1, 5, 8, 12, 14, 11, 5, 1, 8, 13, 20, 26, 25, 16, 6, 1, 13, 21, 33, 46, 51, 41, 22, 7, 1, 21, 34, 54, 79, 97, 92, 63, 29, 8, 1, 34, 55, 88, 133, 176, 189, 155, 92, 37, 9, 1, 55, 89, 143, 221, 309, 365, 344, 247, 129, 46, 10, 1
Offset: 0
Examples
[0] 1; [1] 0, 1; [2] 1, 1, 1; [3] 1, 2, 2, 1; [4] 2, 3, 4, 3, 1; [5] 3, 5, 7, 7, 4, 1; [6] 5, 8, 12, 14, 11, 5, 1; [7] 8, 13, 20, 26, 25, 16, 6, 1; [8] 13, 21, 33, 46, 51, 41, 22, 7, 1; [9] 21, 34, 54, 79, 97, 92, 63, 29, 8, 1;
Crossrefs
Programs
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Maple
T := proc(n, k) option remember; if n = k then 1 elif k = n-1 then n-1 elif k = 0 then T(n-1, 1) else T(n-1, k) + T(n-1, k-1) fi end: seq(seq(T(n, k), k = 0..n), n = 0..11);
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Mathematica
T[n_, k_] := Which[n == k, 1, k == n-1, n-1, k == 0, T[n-1, 1], True, T[n-1, k] + T[n-1, k-1]]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 29 2023 *) -
Python
from functools import cache @cache def A354267row(n): if n == 0: return [1] if n == 1: return [0, 1] row = A354267row(n - 1) + [1] s = row[1] for k in range(n-1, 0, -1): row[k] += row[k - 1] row[0] = s return row for n in range(10): print(A354267row(n))
Formula
T(n, 0) = Fibonacci(n - 1).