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A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.

%I A297899 #18 Jan 20 2024 16:06:54
%S A297899 1,5,1,45,10,1,505,115,15,1,6345,1460,210,20,1,85405,19765,2990,330,
%T A297899 25,1,1204245,279710,43635,5220,475,30,1,17558705,4088615,651165,
%U A297899 81955,8275,645,35,1,262577745,61254760,9901860,1290520,139350,12280,840,40,1
%N A297899 Triangle read by rows, T(n, k) = binomial(n, k)*hypergeom([k-n, n+1], [k+2], -4), for n >= 0 and 0 <= k <= n.
%H A297899 Andrew Howroyd, <a href="/A297899/b297899.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%F A297899 T(n, k) = Sum_{j = k..n} 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1). - _Detlef Meya_, Jan 15 2024
%e A297899 Triangle starts:
%e A297899 [0]       1
%e A297899 [1]       5,      1
%e A297899 [2]      45,     10,     1
%e A297899 [3]     505,    115,    15,    1
%e A297899 [4]    6345,   1460,   210,   20,   1
%e A297899 [5]   85405,  19765,  2990,  330,  25,  1
%e A297899 [6] 1204245, 279710, 43635, 5220, 475, 30, 1
%t A297899 T[n_, k_] := Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 2, -4];
%t A297899 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten
%t A297899 T[n_, k_] := Sum[4^(j - k)*(k + 1)*Binomial[n + j - k, 2*j - k]*Binomial[2*j - k, j - k]/(j + 1), {j, k, n}];
%t A297899 Flatten[Table[T[n, k], {n, 0, 8}, {k, 0, n}]] (* _Detlef Meya_, Jan 15 2024 *)
%o A297899 (PARI) T(n,k) = sum(j = k, n, 4^(j - k)*(k + 1)*binomial(n + j - k, 2*j - k)* binomial(2*j - k, j - k)/(j + 1)) \\ _Andrew Howroyd_, Jan 15 2024
%Y A297899 T(n, 0) = A133305(n). Row sums are A297705, alternating row sums are A131765.
%Y A297899 Cf. A103209.
%K A297899 nonn,tabl
%O A297899 0,2
%A A297899 _Peter Luschny_, Jan 08 2018