A297898 - OEIS

%I #14 Jan 14 2024 12:38:23

%S 1,3,1,13,4,1,63,19,5,1,321,96,26,6,1,1683,501,138,34,7,1,8989,2668,

%T 743,190,43,8,1,48639,14407,4043,1059,253,53,9,1,265729,78592,22180,

%U 5908,1462,328,64,10,1,1462563,432073,122468,33028,8378,1966,416,76,11,1

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

%F T(n, k) = Sum_{j=0..n - k} binomial(n - k, j)*binomial(n + j, j). - _Detlef Meya_, Jan 14 2024

%e Triangle starts:

%e [0]    1

%e [1]    3,    1

%e [2]   13,    4,   1

%e [3]   63,   19,   5,   1

%e [4]  321,   96,  26,   6,  1

%e [5] 1683,  501, 138,  34,  7, 1

%e [6] 8989, 2668, 743, 190, 43, 8, 1

%t T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 1, 2];

%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten

%t T[n_, k_] := Sum[Binomial[n - k, j]*Binomial[n + j, j], {j, 0, n - k}]; Flatten[Table[T[n, k], {n, 0, 9}, {k, 0, n}]] (* _Detlef Meya_, Jan 14 2024 *)

%Y T(n, 0) = A001850(n).

%Y Row sums are A050146(n+1).

%K nonn,tabl

%O 0,2

%A _Peter Luschny_, Jan 08 2018