cp's OEIS Frontend

A297898 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.

%I A297898 #14 Jan 14 2024 12:38:23
%S A297898 1,3,1,13,4,1,63,19,5,1,321,96,26,6,1,1683,501,138,34,7,1,8989,2668,
%T A297898 743,190,43,8,1,48639,14407,4043,1059,253,53,9,1,265729,78592,22180,
%U A297898 5908,1462,328,64,10,1,1462563,432073,122468,33028,8378,1966,416,76,11,1
%N A297898 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 A297898 T(n, k) = Sum_{j=0..n - k} binomial(n - k, j)*binomial(n + j, j). - _Detlef Meya_, Jan 14 2024
%e A297898 Triangle starts:
%e A297898 [0]    1
%e A297898 [1]    3,    1
%e A297898 [2]   13,    4,   1
%e A297898 [3]   63,   19,   5,   1
%e A297898 [4]  321,   96,  26,   6,  1
%e A297898 [5] 1683,  501, 138,  34,  7, 1
%e A297898 [6] 8989, 2668, 743, 190, 43, 8, 1
%t A297898 T[n_, k_] := (-1)^(n - k) Binomial[n, k] Hypergeometric2F1[k - n, n + 1, k + 1, 2];
%t A297898 Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten
%t A297898 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 A297898 T(n, 0) = A001850(n).
%Y A297898 Row sums are A050146(n+1).
%K A297898 nonn,tabl
%O A297898 0,2
%A A297898 _Peter Luschny_, Jan 08 2018