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A124725
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Triangle read by rows: T(n,k) = binomial(n,k) + binomial(n,k+2) (0 <= k <= n).
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2
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1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 7, 8, 7, 4, 1, 11, 15, 15, 11, 5, 1, 16, 26, 30, 26, 16, 6, 1, 22, 42, 56, 56, 42, 22, 7, 1, 29, 64, 98, 112, 98, 64, 29, 8, 1, 37, 93, 162, 210, 210, 162, 93, 37, 9, 1, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10, 1, 56, 176, 385, 627, 792, 792, 627
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OFFSET
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0,4
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COMMENTS
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Binomial transform of the infinite tridiagonal matrix with main diagonal, (1,1,1,...), subdiagonal, (0,0,0,...) and subsubdiagonal, (1,1,1,...). Sum of entries in row n = 2^(n+1) - n - 1 = A000325(n+1).
Riordan array ((1-2x+2x^2)/(1-x)^3, x/(1-x)). - Paul Barry, Apr 08 2011
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LINKS
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FORMULA
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T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k > n. - Philippe Deléham, Feb 12 2014
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EXAMPLE
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Row 3 = (4, 4, 3, 1), then row 4 = (7, 8, 7, 4, 1).
First few rows of the triangle are
1;
1, 1;
2, 2, 1;
4, 4, 3, 1;
7, 8, 7, 4, 1;
11, 15, 15, 11, 5, 1;
16, 26, 30, 26, 16, 6, 1;
...
Production matrix begins
1, 1;
1, 1, 1;
0, 0, 1, 1;
-1, 0, 0, 1, 1;
0, 0, 0, 0, 1, 1;
1, 0, 0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 0, 1, 1;
-1, 0, 0, 0, 0, 0, 0, 1, 1;
0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
(End)
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MAPLE
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T:=(n, k)->binomial(n, k)+binomial(n, k+2): for n from 0 to 12 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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MATHEMATICA
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Flatten[Table[Binomial[n, k]+Binomial[n, k+2], {n, 0, 20}, {k, 0, n}]] (* Harvey P. Dale, Jun 12 2015 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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