A354267 - OEIS

%I #16 Apr 25 2024 12:27:26

%S 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,

%T 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,

%U 55,88,133,176,189,155,92,37,9,1,55,89,143,221,309,365,344,247,129,46,10,1

%N 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.

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F T(n, 0) = Fibonacci(n - 1).

%e [0]  1;

%e [1]  0,  1;

%e [2]  1,  1,  1;

%e [3]  1,  2,  2,  1;

%e [4]  2,  3,  4,  3,  1;

%e [5]  3,  5,  7,  7,  4,  1;

%e [6]  5,  8, 12, 14, 11,  5,  1;

%e [7]  8, 13, 20, 26, 25, 16,  6,  1;

%e [8] 13, 21, 33, 46, 51, 41, 22,  7, 1;

%e [9] 21, 34, 54, 79, 97, 92, 63, 29, 8, 1;

%p T := proc(n, k) option remember;

%p if n = k then 1 elif k = n-1 then n-1 elif k = 0 then T(n-1, 1) else

%p T(n-1, k) + T(n-1, k-1) fi end: seq(seq(T(n, k), k = 0..n), n = 0..11);

%t 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]];

%t Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Nov 29 2023 *)

%o (Python) from functools import cache

%o @cache

%o def A354267row(n):

%o     if n == 0: return [1]

%o     if n == 1: return [0, 1]

%o     row = A354267row(n - 1) + [1]

%o     s = row[1]

%o     for k in range(n-1, 0, -1):

%o         row[k] += row[k - 1]

%o     row[0] = s

%o     return row

%o for n in range(10): print(A354267row(n))

%Y Cf. A212804 (first column, which is also row 0 of A352744), A099036 (row sums), A228074 (subtriangle), A000045 (Fibonacci), A371870 (central terms).

%K nonn,tabl

%O 0,8

%A _Peter Luschny_, May 31 2022