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A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).

%I A184883 #10 Sep 08 2022 08:45:55
%S A184883 1,1,1,1,5,1,1,9,13,1,1,13,41,25,1,1,17,85,129,41,1,1,21,145,377,321,
%T A184883 61,1,1,25,221,833,1289,681,85,1,1,29,313,1561,3649,3653,1289,113,1,1,
%U A184883 33,421,2625,8361,13073,8989,2241,145,1,1,37,545,4089,16641,36365,40081,19825,3649,181,1
%N A184883 Number triangle T(n,k) = [k<=n]*Hypergeometric2F1([-k, 2k-2n], [1], 2).
%H A184883 G. C. Greubel, <a href="/A184883/b184883.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A184883 T(n,k) = [k<=n]*Sum_{j=0..k} C(2*n-2*k,j)*C(k,j)*2^j.
%F A184883 T(n, n-k) = A114123(n, k).
%F A184883 Sum_{k=0..n} T(n, k) = A099463(n+1).
%F A184883 Sum_{k=0..floor(n/2)} T(n, k) = A184884(n).
%F A184883 T(n, k) = Hypergeometric2F1([-k, 2*(k-n)], [1], 2). - _G. C. Greubel_, Nov 19 2021
%e A184883 Triangle begins
%e A184883   1;
%e A184883   1,  1;
%e A184883   1,  5,   1;
%e A184883   1,  9,  13,    1;
%e A184883   1, 13,  41,   25,     1;
%e A184883   1, 17,  85,  129,    41,     1;
%e A184883   1, 21, 145,  377,   321,    61,     1;
%e A184883   1, 25, 221,  833,  1289,   681,    85,     1;
%e A184883   1, 29, 313, 1561,  3649,  3653,  1289,   113,    1;
%e A184883   1, 33, 421, 2625,  8361, 13073,  8989,  2241,  145,   1;
%e A184883   1, 37, 545, 4089, 16641, 36365, 40081, 19825, 3649, 181, 1;
%t A184883 A184883[n_, k_]:= Hypergeometric2F1[-k, 2*(k-n), 1, 2];
%t A184883 Table[A184883[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Nov 19 2021 *)
%o A184883 (Magma)
%o A184883 T:= func< n,k | (&+[Binomial(k,j)*Binomial(2*(n-k), j)*2^j: j in [0..k]]) >;
%o A184883 [T(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Nov 19 2021
%o A184883 (Sage)
%o A184883 def A184883(n,k): return simplify( hypergeometric([-k, 2*(k-n)], [1], 2) )
%o A184883 flatten([[A184883(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Nov 19 2021
%Y A184883 Cf. A099463 (row sums), A114123, A184884 (diagonal sums).
%K A184883 nonn,easy,tabl
%O A184883 0,5
%A A184883 _Paul Barry_, Jan 24 2011