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A071675
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Array read by antidiagonals of trinomial coefficients.
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19
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1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 0, 3, 3, 1, 0, 0, 2, 6, 4, 1, 0, 0, 1, 7, 10, 5, 1, 0, 0, 0, 6, 16, 15, 6, 1, 0, 0, 0, 3, 19, 30, 21, 7, 1, 0, 0, 0, 1, 16, 45, 50, 28, 8, 1, 0, 0, 0, 0, 10, 51, 90, 77, 36, 9, 1, 0, 0, 0, 0, 4, 45, 126, 161, 112, 45, 10, 1, 0, 0, 0, 0, 1, 30, 141, 266, 266
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OFFSET
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0,9
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COMMENTS
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Read as a number triangle, this is the Riordan array (1, x(1+x+x^2)) with T(n,k) = Sum_{i=0..floor((n+k)/2)} C(k,2i+2k-n)*C(2i+2k-n,i). Rows start {1}, {0,1}, {0,1,1}, {0,1,2,1}, {0,0,3,3,1},... Row sums are then the trinomial numbers A000073(n+2). Diagonal sums are A013979. - Paul Barry, Feb 15 2005
Let {a_(k,i)}, k>=1, i=0,...,k, be the k-th antidiagonal of the array. Then s_k(n)=sum{i=0,...,k}a_(k,i)* binomial(n,k) is the n-th element of the k-th column of A213742. For example, s_1(n)=binomial(n,1)=n is the first column of A213742 for n>1, s_2(n)=binomial(n,1)+binomial(n,2)is the second column of A213742 for n>1, etc. In particular (see comment in A213742) in cases k=4,5,6,7,8, s_k(n) is A005718(n+2), A005719(n), A005720(n), A001919(n), A064055(n+3), respectively. - Vladimir Shevelev and Peter J. C. Moses, Jun 22 2012
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LINKS
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FORMULA
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T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-1, k-2) starting with T(0, 0)=1. See A027907 for more.
As a number triangle, T(n, k) = Sum_{i=0..floor((n-k)/2)} C(n-k-i, i) * C(k, n-k-i). - Paul Barry, Apr 26 2005
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EXAMPLE
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Rows start
1, 0, 0, 0, 0, 0, ...;
1, 1, 1, 0, 0, 0, 0, ...;
1, 2, 3, 2, 1, 0, 0, ...;
1, 3, 6, 7, 6, 3, 1, 0, ...;
1, 4, 10, 16, 19, 16, 10, 4, 1, ...; etc.
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MATHEMATICA
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T[n_, k_] := Sum[Binomial[n - k - j, j]*Binomial[k, n - k - j], {j, 0,
Floor[(n - k)/2]}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Feb 28 2017 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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