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A121574
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Riordan array (1/(1-2*x), x*(1+x)/(1-2*x)).
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2
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1, 2, 1, 4, 5, 1, 8, 16, 8, 1, 16, 44, 37, 11, 1, 32, 112, 134, 67, 14, 1, 64, 272, 424, 305, 106, 17, 1, 128, 640, 1232, 1168, 584, 154, 20, 1, 256, 1472, 3376, 3992, 2641, 998, 211, 23, 1, 512, 3328, 8864, 12592, 10442, 5221, 1574, 277, 26, 1
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OFFSET
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0,2
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COMMENTS
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A generalized Delannoy number triangle.
The following remarks assume the row indexing starts at n = 1.
The sequence of row polynomials R(n,x), beginning R(1,x) = 1, R(2,x) = 2 + x, R(3,x) = 4 + 5*x + x^2 , ..., is a strong divisibility sequence of polynomials in the ring Z[x]; that is, for all positive integers n and m, poly_gcd( R(n,x), R(m,x)) = R(gcd(n, m), x) - apply Norfleet (2005), Theorem 3. Consequently, the polynomial sequence {R(n,x): n >= 1} is a divisibility sequence; that is, if n divides m then R(n,x) divides R(m,x) in Z[x]. (End)
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LINKS
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FORMULA
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Number array T(n,k) = Sum_{j=0..n-k} C(k,j)*C(n-j,k)*2^(n-k-j).
Recurrence for row polynomials (with row indexing starting at n = 1): R(n,x) = (x + 2)*R(n-1,x) + x*R(n-2,x) with R(1,x) = 1 and R(2,x) = x + 2. - Peter Bala, Feb 07 2024
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EXAMPLE
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Triangle begins
1;
2, 1;
4, 5, 1;
8, 16, 8, 1;
16, 44, 37, 11, 1;
32, 112, 134, 67, 14, 1;
64, 272, 424, 305, 106, 17, 1;
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MAPLE
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T:=(n, k)->add(binomial(k, j)*binomial(n-j, k)*2^(n-k-j), j=0..n-k): seq(seq(T(n, k), k=0..n), n=0..9); # Muniru A Asiru, Nov 02 2018
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MATHEMATICA
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Table[Sum[Binomial[k, j] Binomial[n-j, k] 2^(n-k-j), {j, 0, n-k}], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Nov 02 2018 *)
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PROG
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(PARI) for(n=0, 10, for(k=0, n, print1(sum(j=0, n-k, binomial(k, j)* binomial(n-j, k)*2^(n-k-j)), ", "))) \\ G. C. Greubel, Nov 02 2018
(Magma) [[(&+[ Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j): j in [0..(n-k)]]): k in [0..n]]: n in [0..10]]; // G. C. Greubel, Nov 02 2018
(GAP) T:=Flat(List([0..9], n->List([0..n], k->Sum([0..n-k], j->Binomial(k, j)*Binomial(n-j, k)*2^(n-k-j))))); # Muniru A Asiru, Nov 02 2018
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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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