A193922 - OEIS

%I #22 Jan 12 2024 10:04:27

%S 1,1,1,2,2,1,4,4,3,2,7,7,6,5,3,12,12,11,10,8,5,20,20,19,18,16,13,8,33,

%T 33,32,31,29,26,21,13,54,54,53,52,50,47,42,34,21,88,88,87,86,84,81,76,

%U 68,55,34,143,143,142,141,139,136,131,123,110,89,55,232,232,231

%N Mirror of the triangle A193921.

%C A193922 is obtained by reversing the rows of the triangle A193921.

%C Also, triangle read by rows: T(n,k) = Fibonacci(n+2) - Fibonacci(k+1) with T(0,0) = 1, 0 <= k <= n. - _Arkadiusz Wesolowski_, Aug 05 2012

%H Arkadiusz Wesolowski, <a href="/A193922/b193922.txt">Rows n = 0..140 of triangle, flattened</a>

%F Write w(n,k) for the triangle at A193921.  The triangle at A193922 is then given by w(n,n-k).

%F G.f.: 1-(x*y-y-x)/((x^2+x-1)*(y^2+y-1)). - _Vladimir Kruchinin_, Jan 12 2024

%e First six rows:

%e    1

%e    1   1

%e    2   2   1

%e    4   4   3   2

%e    7   7   6   5   3

%e   12  12  11  10   8   5

%t z = 12;

%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := x*q[n - 1, x] + 1; q[0, n_] := 1;

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193921 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]]  (* this sequence *)

%t Factor[w[7, x]]

%t Factor[w[8, x]]

%t Table[Expand[p[n, x]], {n, 0, 4}]

%t Table[Expand[q[n, x]], {n, 0, 4}]

%t Prepend[Flatten@Table[Fibonacci[n + 2] - Fibonacci[k + 1], {n, 10}, {k, 0, n}], 1] (* _Arkadiusz Wesolowski_, Aug 05 2012 *)

%Y Cf. A193921.

%K nonn,tabl

%O 0,4

%A _Clark Kimberling_, Aug 09 2011