OFFSET
1,5
COMMENTS
Are there closed forms for diagonals and columns?
With the definition of the array, I note that the sequence (phi(k)) (phi(k)= g.f. of the column number k) is given by the recurrence relation: phi(k+1)=phi(k)*(1-z^k)/(1-2*z+z^(k+1)). The consequence is: the sequence number k+1 column is the convolution of the k-one and a "-acci like" sequence whose g.f. is given by (1-z^k)/(1-2*z+z^(k+1)). E.g., the 2-column is the convolution of the 1-column and the sequence 1, 2, 3, 5, ... classical Fibonacci sequence without the first 1. The 3-column is the convolution of the 2-column and 1, 2, 4, 7, 13, ... tribonacci like-sequence (exactly: A000073 without beginning 0, 0, 1). - Richard Choulet, Feb 19 2010
Relation to metallic means:
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 0*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the golden ratio, A001622.
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 1*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the silver ratio, A014176.
T(n,1)=1, k>1: T(n,k) = Sum_{i=1..k-1} T(n-i,k-1) + 2*Sum_{i=1..k-1} T(n-i,k)
has antidiagonal sums for which the limiting ratio tends to the bronze ratio, A098316.
A similar point can be made about variations of the Pascal triangle.
LINKS
Wikipedia, Metallic mean
FORMULA
The g.f of the number k column is phi(k)(z) = (1/(1-z))*Product_{i=1..k-1}(1-z^i)/(1-2*z+z^(i+1)). - Richard Choulet, Feb 19 2010
EXAMPLE
Table begins:
n/k| 1 2 3 4 5 6 7 8 9 10 11
---+-----------------------------------------------------
1 | 1
2 | 1 1
3 | 1 2 1
4 | 1 3 4 1
5 | 1 4 10 6 1
6 | 1 5 21 22 8 1
7 | 1 6 40 64 38 10 1
8 | 1 7 72 163 140 58 12 1
9 | 1 8 125 382 442 256 82 14 1
10 | 1 9 212 846 1259 954 420 110 16 1
11 | 1 10 354 1800 3334 3166 1794 640 142 18 1
Example: T(8,4) = 163 because it is the sum of the numbers:
10 6
21 22
40 64
For k=1, we obtain phi(k)(z)=1/(1-z) which is clear; for k=2, we obtain phi(k)(z)=1/(1-z)^2. For k=3, we obtain phi(3)(z)=(1+z)/((1-2*z+z^3)*(1-z)); this is A001891 without the beginning zero. - Richard Choulet, Feb 19 2010
MAPLE
A175105 := proc(n, k) if k =1 then 1; elif k > n or k< 1 then 0 ; else add(procname(n-i, k-1)+procname(n-i, k), i=1..k-1) ; end if; end proc; # R. J. Mathar, Feb 16 2011
MATHEMATICA
T[_, 1] = 1;
T[n_, k_] /; 1<k<=n := T[n, k] = Sum[T[n-i, k-1]+T[n-i, k], {i, 1, k-1}];
T[_, _] = 0;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 19 2019 *)
PROG
(Excel) =if(column()=1; 1; if(row()>=column(); sum(indirect(address(row()-column()+1; column()-1; 4)&":"&address(row()-column()+column()-1; column()-1; 4); 4))+sum(indirect(address(row()-column()+1; column(); 4)&":"&address(row()-column()+column()-1; column(); 4); 4)); 0)) ' Mats Granvik, Mar 28 2010
CROSSREFS
(1-((-1)^T(n, k)))/2 = T(n, k) mod 2 = A051731.
KEYWORD
nonn,tabl
AUTHOR
Mats Granvik, Feb 10 2010
EXTENSIONS
Corrected and edited by Mats Granvik, Jul 28 2010, Dec 09 2010
Choulet formulas indices shifted (to adapt to the new column index) by R. J. Mathar, Dec 13 2010
STATUS
approved