cp's OEIS Frontend

A349727 Triangle read by rows, T(n, k) = [x^(n - k)] hypergeom([-n, -1 + n], [-n], x).

%I A349727 #17 Jul 31 2023 21:31:24
%S A349727 1,0,1,1,1,1,4,3,2,1,15,10,6,3,1,56,35,20,10,4,1,210,126,70,35,15,5,1,
%T A349727 792,462,252,126,56,21,6,1,3003,1716,924,462,210,84,28,7,1,11440,6435,
%U A349727 3432,1716,792,330,120,36,8,1,43758,24310,12870,6435,3003,1287,495,165,45,9,1
%N A349727 Triangle read by rows, T(n, k) = [x^(n - k)] hypergeom([-n, -1 + n], [-n], x).
%e A349727 Triangle starts:
%e A349727 [0] 1;
%e A349727 [1] 0,     1;
%e A349727 [2] 1,     1,    1;
%e A349727 [3] 4,     3,    2,    1;
%e A349727 [4] 15,    10,   6,    3,    1;
%e A349727 [5] 56,    35,   20,   10,   4,   1;
%e A349727 [6] 210,   126,  70,   35,   15,  5,   1;
%e A349727 [7] 792,   462,  252,  126,  56,  21,  6,   1;
%e A349727 [8] 3003,  1716, 924,  462,  210, 84,  28,  7,  1;
%e A349727 [9] 11440, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1;
%p A349727 p := n -> hypergeom([-n, -1 + n], [-n], x):
%p A349727 seq(seq(coeff(simplify(p(n)), x, n - k), k = 0..n), n = 0..10);
%t A349727 (* rows[0..k], k[0..oo] *)
%t A349727 r={};k=11;For[n=0,n<k+1,n++,AppendTo[r,Binomial[2*k-2-n,k-2]]];r
%t A349727 (* columns [0..n], n[0..oo] *)
%t A349727 c={};n=0;For[k=n,k<n+13,k++,AppendTo[c,Binomial[2*k-2-n,k-2]]];c
%t A349727 (* sequence *)
%t A349727 s={};For[k=0,k<13,k++,For[n=0,n<k+1,n++,AppendTo[s,Binomial[2*k-2-n,k-2]]]];s (* _Detlef Meya_, Jun 26 2023 *)
%Y A349727 Row sums: A088218, alternating row sums: A091526.
%Y A349727 Central coefficients: binomial(3*n-2, n) (cf. A117671).
%Y A349727 T(n, 0) = binomial(2*(n-1), n) (cf. A001791).
%Y A349727 Cf. A257635.
%K A349727 nonn,tabl
%O A349727 0,7
%A A349727 _Peter Luschny_, Nov 27 2021