A193913 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).

1, 3, 15, 79, 425, 2317, 12749, 70631, 393379, 2200203, 12348645, 69507969, 392211153, 2217824883, 12564291759, 71294454543, 405135974649, 2305189276605, 13131574749357, 74883034577575, 427430124521651, 2441889639394043, 13961588736578245, 79884779408549249

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1309

Formula

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

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

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

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

T(n,2) = A001911(n). T(n,3) = A000045(n+6) - A004767(n+1). - R. J. Mathar, Aug 28 2011

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

Example

Diagonal of the matrix T(n,k) which starts for n,k >= 1 as:
1    1   2    3    5     8    13    21     34     55
1    3   6   11   19    32    53    87    142    231
2    6  15   32   62   113   198   338    567    940
3   11  32   79  173   348   659  1195   2100   3607
5   19  62  173  425   946  1953  3807   7102  12809
8   32 113  348  946  2317  5216 10976  21885  41796
13  53 198  659 1953  5216 12749 28941  61802 125483
21  87 338 1195 3807 10976 28941 70631 161374 348659

Maple

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:
A193913 := proc(n) A(n, n) ; end proc: # R. J. Mathar, Aug 28 2011
# second Maple program:
b:= proc(x, y) option remember; `if`(x<2, (<<0|1>, <1|1>>^y)[1, 2],
      b(x-1, y)+b(sort([x, y-1])[])+b(x-1, y-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=1..29);  # Alois P. Heinz, Jul 14 2021

Mathematica

T[n_ /; n>=1, 1] := T[1, n] = Fibonacci[n];
T[n_ /; n>=1, k_] /; n>=k := T[n, k] = T[n-1, k-1] + T[n, k-1] + T[n-1, k];
T[n_, k_] /; k>n := T[k, n];
T[_, _] = 0;
a[n_] := T[n, n];
Array[a, 24] (* Jean-François Alcover, Apr 26 2022 *)

Prog

(MATLAB) function [ out ] = a( n )
ary=zeros(n, n);
ary(1, 1)=1;
if(n==1)
out= 1;
return;
end
ary(2, 1)=1;
ary(1, 2)=1;
for i=3:n
ary(i, 1)=ary(i-1, 1)+ary(i-2, 1);
ary(1, i)=ary(1, i-1)+ary(1, i-2);
end
for i=2:n
for j=2:n
ary(i, j)=ary(i, j-1)+ary(i-1, j-1)+ary(i-1, j);
end
end
out=ary(n, n)

Crossrefs

Cf. A000045.

Cf. A001911, A004767.

Sequence in context: A031884 A337284 A186942 * A292097 A052755 A137957

Adjacent sequences: A193910 A193911 A193912 * A193914 A193915 A193916

Keyword

nonn

Author

Jaley H Dholakiya, Aug 09 2011

Status

approved