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A114123
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Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).
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6
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1, 1, 1, 1, 5, 1, 1, 13, 9, 1, 1, 25, 41, 13, 1, 1, 41, 129, 85, 17, 1, 1, 61, 321, 377, 145, 21, 1, 1, 85, 681, 1289, 833, 221, 25, 1, 1, 113, 1289, 3653, 3649, 1561, 313, 29, 1, 1, 145, 2241, 8989, 13073, 8361, 2625, 421, 33, 1, 1, 181, 3649, 19825, 40081, 36365, 16641, 4089, 545, 37, 1
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OFFSET
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0,5
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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(2*k,n-k-j)*C(n-k,j)*2^(n-k-j).
T(n, k) = Sum_{j=0..n-k} C(2*k,j)*C(n-k,j)*2^j.
Sum_{k=0..n} T(n, k) = A099463(n+1).
Sum_{k=0..floor(n/2)} T(n, k) = A116404(n).
T(n, k) = hypergeom([-2*k, k-n], [1], 2). - Peter Luschny, Sep 16 2014
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EXAMPLE
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Triangle begins
1;
1, 1;
1, 5, 1;
1, 13, 9, 1;
1, 25, 41, 13, 1;
1, 41, 129, 85, 17, 1;
1, 61, 321, 377, 145, 21, 1;
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MAPLE
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T := (n, k) -> hypergeom([-2*k, k-n], [1], 2);
seq(seq(round(evalf(T(n, k), 99)), k=0..n), n=0..9); # Peter Luschny, Sep 16 2014
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MATHEMATICA
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T[n_, k_] := Hypergeometric2F1[-2k, k-n, 1, 2];
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PROG
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(Magma)
T:= func< n, k | (&+[Binomial(2*k, j)*Binomial(n-k, j)*2^j: j in [0..n-k]]) >;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Nov 20 2021
(Sage)
def A114123(n, k): return round( hypergeometric([-2*k, k-n], [1], 2) )
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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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