A111669 - OEIS

%I #30 May 25 2024 09:05:31

%S 1,1,1,1,2,1,1,3,4,1,1,4,11,7,1,1,5,26,32,12,1,1,6,57,122,92,20,1,1,7,

%T 120,423,582,252,33,1,1,8,247,1389,3333,2598,681,54,1,1,9,502,4414,

%U 18054,24117,11451,1815,88,1,1,10,1013,13744,94684,210990,172980,49566,4807,143,1

%N Triangle read by rows, based on a simple Fibonacci recursion rule.

%C Subdiagonal is A000071(n+3). Row sums of inverse are 0^n.

%C Row sums are given by A135934. - _Emanuele Munarini_, Dec 05 2017

%H Michel Marcus, <a href="/A111669/b111669.txt">Rows n = 0..20, flattened</a>

%F T(n, k) = T(n-1, k-1) + F(k+1)*T(n-1, k) where F(n)=A000045(n).

%F Column k has g.f. x^k/Product_{j=0..k} (1 - F(j+1)*x).

%e Triangle begins

%e   1....1....2....3....5....8...13....F(k+1)

%e   1

%e   1....1

%e   1....2....1

%e   1....3....4....1

%e   1....4...11....7....1

%e   1....5...26...32...12....1

%e   1....6...57..122...92...20....1

%e For example, T(6,3) = 122 = 26 + 3*32 = T(5,2) + F(4)*T(5,3).

%t (* To generate the triangle *)

%t Grid[RecurrenceTable[{F[n,k] == F[n-1,k-1] + Fibonacci[k+1] F[n-1,k], F[0,k] == KroneckerDelta[k]}, F, {n,0,10}, {k,0,10}]] (* _Emanuele Munarini_, Dec 05 2017 *)

%o (PARI) T(n, k) = if ((n<0) || (k<0), 0, if ((n==0) && (k==0), 1, T(n-1, k-1) + fibonacci(k+1)*T(n-1, k))); \\ _Michel Marcus_, May 25 2024

%Y Cf. A111577, A111578, A111579, A008277, A039755, A135934.

%K easy,nonn,tabl

%O 0,5

%A _Gary W. Adamson_, Aug 14 2005

%E Edited by _Paul Barry_, Nov 14 2005