A175105 Triangle T(n,k) read by rows. T(n,1)=1; T(n,k) = Sum_{i=1..k-1} ( T(n-i,k-1) + T(n-i,k) ), k>1.

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 10, 6, 1, 1, 5, 21, 22, 8, 1, 1, 6, 40, 64, 38, 10, 1, 1, 7, 72, 163, 140, 58, 12, 1, 1, 8, 125, 382, 442, 256, 82, 14, 1, 1, 9, 212, 846, 1259, 954, 420, 110, 16, 1, 1, 10, 354, 1800, 3334, 3166, 1794, 640, 142, 18, 1, 1, 11, 585, 3719, 8366, 9657, 6754, 3074, 924, 178, 20, 1

Offset: 1

Author

Mats Granvik, Feb 10 2010

Keywords

• nonn
• tabl

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.

Examples

			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
		

Links

• Wikipedia, Metallic mean

Crossrefs

Cf. A172119, A051731, A001891 (column k=3), A176084 (row sums).

(1-((-1)^T(n, k)))/2 = T(n, k) mod 2 = A051731.

Cf. A179807=antidiagonal sums. A179748 has simpler recurrence.

Programs

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

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, ] = 0;
Table[T[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Dec 19 2019 *)

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

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