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A175105
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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.
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8
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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
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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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
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MATHEMATICA
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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;
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PROG
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(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
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CROSSREFS
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(1-((-1)^T(n, k)))/2 = T(n, k) mod 2 = A051731.
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KEYWORD
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AUTHOR
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EXTENSIONS
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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
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STATUS
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approved
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