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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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 silver 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.

Closed form for diagonals. The first two are banal: a1(n) = 1 and a2(n) = 2(n-1) for n>=2. Then if we calculate the subsequent adjacent terms' differences in any other diagonals we see that they reach a constant value. For the k-th diagonal after k differences we reach the constant value of 2^(k-1). In any case, the closed form is a polynomial of degree (k-1), apart from some initial perturbation due to the fact that in the first terms of any diagonals some zeros are added. Then the closed form for the third diagonal (1, 3, 10, 22, 38, ...) is a3(n) = 2*(n^2-n-1) with n>=3. For the fourth (1, 4, 21, 64, 140, ...) it is a4(n) = (4/3)*n*(n^2 - 4) with n>=4. For the fifth (1, 5, 40, 163, 442, ...) it is a5(n) = (2/3)*(n^4 + 2*n^3 - 7*n^2 - 8*n + 3) for n>=5 and so on. - Paolo P. Lava, Mar 22 2011

LINKS

Table of n, a(n) for n=1..71.

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

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

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.

Sequence in context: A144823 A098446 A098447 * A162717 A122175 A073165

Adjacent sequences:  A175102 A175103 A175104 * A175106 A175107 A175108

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

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Last modified November 21 12:34 EST 2017. Contains 295001 sequences.