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A114123 Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).

%I A114123 #23 Sep 08 2022 08:45:23
%S A114123 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,
%T A114123 21,1,1,85,681,1289,833,221,25,1,1,113,1289,3653,3649,1561,313,29,1,1,
%U A114123 145,2241,8989,13073,8361,2625,421,33,1,1,181,3649,19825,40081,36365,16641,4089,545,37,1
%N A114123 Riordan array (1/(1-x), x*(1+x)^2/(1-x)^2).
%C A114123 Row sums are A099463(n+1). Diagonal sums are A116404.
%C A114123 Triangle formed of even-numbered columns of the Delannoy triangle A008288. - _Philippe Deléham_, Mar 11 2013
%H A114123 G. C. Greubel, <a href="/A114123/b114123.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A114123 T(n, k) = Sum_{j=0..n} C(2*k,n-k-j)*C(n-k,j)*2^(n-k-j).
%F A114123 T(n, k) = Sum_{j=0..n-k} C(2*k,j)*C(n-k,j)*2^j.
%F A114123 Sum_{k=0..n} T(n, k) = A099463(n+1).
%F A114123 Sum_{k=0..floor(n/2)} T(n, k) = A116404(n).
%F A114123 T(n, k) = hypergeom([-2*k, k-n], [1], 2). - _Peter Luschny_, Sep 16 2014
%F A114123 T(n, n-k) = A184883(n, k). - _G. C. Greubel_, Nov 20 2021
%e A114123 Triangle begins
%e A114123   1;
%e A114123   1,  1;
%e A114123   1,  5,   1;
%e A114123   1, 13,   9,   1;
%e A114123   1, 25,  41,  13,   1;
%e A114123   1, 41, 129,  85,  17,  1;
%e A114123   1, 61, 321, 377, 145, 21, 1;
%p A114123 T := (n,k) -> hypergeom([-2*k, k-n], [1], 2);
%p A114123 seq(seq(round(evalf(T(n,k),99)),k=0..n),n=0..9); # _Peter Luschny_, Sep 16 2014
%t A114123 T[n_, k_] := Hypergeometric2F1[-2k, k-n, 1, 2];
%t A114123 Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* _Jean-François Alcover_, Jun 13 2019 *)
%o A114123 (Magma)
%o A114123 T:= func< n, k | (&+[Binomial(2*k, j)*Binomial(n-k, j)*2^j: j in [0..n-k]]) >;
%o A114123 [T(n, k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 20 2021
%o A114123 (Sage)
%o A114123 def A114123(n,k): return round( hypergeometric([-2*k, k-n], [1], 2) )
%o A114123 flatten([[A114123(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 20 2021
%Y A114123 Cf. A008288, A099463 (row sums), A116404 (diagonal sums), A184883.
%K A114123 easy,nonn,tabl
%O A114123 0,5
%A A114123 _Paul Barry_, Feb 07 2006, Oct 22 2006