A193913 - OEIS

%I #32 Apr 26 2022 10:26:54

%S 1,3,15,79,425,2317,12749,70631,393379,2200203,12348645,69507969,

%T 392211153,2217824883,12564291759,71294454543,405135974649,

%U 2305189276605,13131574749357,74883034577575,427430124521651,2441889639394043,13961588736578245,79884779408549249

%N Diagonal element T(n,n) of the infinite array with T(n,1) = T(1,n) = Fibonacci(n) and recursively T(n,k) = T(n-1,k-1) + T(n,k-1) + T(n-1,k).

%H Alois P. Heinz, <a href="/A193913/b193913.txt">Table of n, a(n) for n = 1..1309</a>

%F T(n,1) = T(1,n) = A000045(n).

%F T(n,k) = 0 if n <= 0 or k <= 0.

%F T(n,k) = T(n-1,k-1) + T(n,k-1) + T(n-1,k), n > 1, k > 1.

%F T(n,k) = T(k,n).

%F T(n,2) = A001911(n). T(n,3) = A000045(n+6) - A004767(n+1). - _R. J. Mathar_, Aug 28 2011

%F T(n,4) = A000045(n+9) - 34 - 14*n - 4*n^2. T(n,5) = A000045(n+12) -8*n^3/3 -14*n^2 -208*n/3 -142. - _R. J. Mathar_, Aug 29 2011

%e Diagonal of the matrix T(n,k) which starts for n,k >= 1 as:

%e 1    1   2    3    5     8    13    21     34     55

%e 1    3   6   11   19    32    53    87    142    231

%e 2    6  15   32   62   113   198   338    567    940

%e 3   11  32   79  173   348   659  1195   2100   3607

%e 5   19  62  173  425   946  1953  3807   7102  12809

%e 8   32 113  348  946  2317  5216 10976  21885  41796

%e 13  53 198  659 1953  5216 12749 28941  61802 125483

%e 21  87 338 1195 3807 10976 28941 70631 161374 348659

%p A := proc(n,k) option remember; if n<=0 or k<=0 then 0; elif k =  1 then combinat[fibonacci](n) ; elif n =  1 then combinat[fibonacci](k) ; else procname(n-1,k-1)+procname(n,k-1)+procname(n-1,k) ; end if; end proc:

%p A193913 := proc(n) A(n,n) ; end proc: # _R. J. Mathar_, Aug 28 2011

%p # second Maple program:

%p b:= proc(x, y) option remember; `if`(x<2, (<<0|1>, <1|1>>^y)[1, 2],

%p       b(x-1, y)+b(sort([x, y-1])[])+b(x-1, y-1))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=1..29);  # _Alois P. Heinz_, Jul 14 2021

%t T[n_ /; n>=1, 1] := T[1, n] = Fibonacci[n];

%t T[n_ /; n>=1, k_] /; n>=k := T[n, k] = T[n-1, k-1] + T[n, k-1] + T[n-1, k];

%t T[n_, k_] /; k>n := T[k, n];

%t T[_, _] = 0;

%t a[n_] := T[n, n];

%t Array[a, 24] (* _Jean-François Alcover_, Apr 26 2022 *)

%o (MATLAB) function [ out ] = a( n )

%o ary=zeros(n,n);

%o ary(1,1)=1;

%o if(n==1)

%o out= 1;

%o return;

%o end

%o ary(2,1)=1;

%o ary(1,2)=1;

%o for i=3:n

%o ary(i,1)=ary(i-1,1)+ary(i-2,1);

%o ary(1,i)=ary(1,i-1)+ary(1,i-2);

%o end

%o for i=2:n

%o for j=2:n

%o ary(i,j)=ary(i,j-1)+ary(i-1,j-1)+ary(i-1,j);

%o end

%o end

%o out=ary(n,n)

%Y Cf. A000045.

%Y Cf. A001911, A004767.

%K nonn

%O 1,2

%A _Jaley H Dholakiya_, Aug 09 2011