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A113139
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Number triangle, equal to half of Delannoy square array A008288.
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8
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1, 3, 1, 13, 5, 1, 63, 25, 7, 1, 321, 129, 41, 9, 1, 1683, 681, 231, 61, 11, 1, 8989, 3653, 1289, 377, 85, 13, 1, 48639, 19825, 7183, 2241, 575, 113, 15, 1, 265729, 108545, 40081, 13073, 3649, 833, 145, 17, 1, 1462563, 598417, 224143, 75517, 22363, 5641
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OFFSET
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0,2
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COMMENTS
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LINKS
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FORMULA
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T(n, k) = Sum_{j=0..n} C(n-k, j)*C(n+j, k+j).
T(n, k) = Sum_{j=0..n} C(n, j)*C(n-k, j-k)*2^(n-j).
O.g.f.: 2/( sqrt(x^2 - 6*x + 1)*(t*sqrt(x^2 - 6*x + 1) + t*x - t + 2) ) = 1 + (3 + t)*x + (13 + 5*t + t^2)*x^2 + ....
Riordan array (f(x), x*g(x)), where f(x) = 1/sqrt(1 - 6*x + x^2) is the o.g.f. for the central Delannoy numbers, A001850, and g(x) = 1/x* revert( x*(1 - x)/(1 + x) ) = 1 + 2*x + 6*x^2 + 22*x^3 + 90*x^4 + 394*x^5 + ... is the o.g.f. for the large Schroder numbers, A006318.
Read as a square array, this is the generalized Riordan array (f(x), g(x)) in the sense of the Bala link, which factorizes as (1 + x*g'(x)/g(x), x*g(x)) * (1/(1 - x), (1 + x)/(1 - x)) = A110171 * A008288. See the example below. (End)
T(n,k) = (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2). - Peter Luschny, Mar 02 2017
T(n,k) = P(n-k, k, 0, 3), where P(n, alpha, beta, x) is the n-th Jacobi polynomial with parameters alpha and beta.
T(n,k) = binomial(n,k) * hypergeom( [n + 1, k - n], [k + 1], -1 ).
The n-th row polynomial in descending powers of x is the n-th Taylor polynomial of the rational function (1 + x)^n/(1 - x)^(n+1) about 0. For example, for n = 4, (1 + x)^4/(1 - x)^5 = 1 + 9*x + 41*x^2 + 129*x^3 + 321*x^4 + O(x^5). Cf. A110171. (End)
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EXAMPLE
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Triangle begins
1;
3, 1;
13, 5, 1;
63, 25, 7, 1;
321, 129, 41, 9, 1;
1683, 681, 231, 61, 11, 1;
8989, 3653, 1289, 377, 85, 13, 1;
...
/ 1 1 1 1 ... \ / 1 \ / 1 1 1 1 ...\
| 3 5 7 9 ... | | 2 1 || 1 3 5 7 ...|
|13 25 41 61 ... | = | 8 4 1 || 1 5 13 25 ...|
|63 129 231 377 ... | |38 18 6 1 || 1 7 25 63 .. |
|... | |... || 1... |
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MAPLE
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T := (n, k) -> (-1)^(n-k)*hypergeom([n+1, -n+k], [1], 2):
seq(seq(simplify(T(n, k)), k=0..n), n=0..8); # Peter Luschny, Mar 02 2017
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MATHEMATICA
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Table[Sum[Binomial[n - k, j] Binomial[n + j, k + j], {j, 0, n}], {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Dec 09 2015 *)
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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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