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Triangle read by rows, T(n, k) = [x^(n - k)] hypergeom([-n, -1 + n], [-n], x).
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%I #17 Jul 31 2023 21:31:24

%S 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 792,462,252,126,56,21,6,1,3003,1716,924,462,210,84,28,7,1,11440,6435,

%U 3432,1716,792,330,120,36,8,1,43758,24310,12870,6435,3003,1287,495,165,45,9,1

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

%e Triangle starts:

%e [0] 1;

%e [1] 0, 1;

%e [2] 1, 1, 1;

%e [3] 4, 3, 2, 1;

%e [4] 15, 10, 6, 3, 1;

%e [5] 56, 35, 20, 10, 4, 1;

%e [6] 210, 126, 70, 35, 15, 5, 1;

%e [7] 792, 462, 252, 126, 56, 21, 6, 1;

%e [8] 3003, 1716, 924, 462, 210, 84, 28, 7, 1;

%e [9] 11440, 6435, 3432, 1716, 792, 330, 120, 36, 8, 1;

%p p := n -> hypergeom([-n, -1 + n], [-n], x):

%p seq(seq(coeff(simplify(p(n)), x, n - k), k = 0..n), n = 0..10);

%t (* rows[0..k], k[0..oo] *)

%t r={};k=11;For[n=0,n<k+1,n++,AppendTo[r,Binomial[2*k-2-n,k-2]]];r

%t (* columns [0..n], n[0..oo] *)

%t c={};n=0;For[k=n,k<n+13,k++,AppendTo[c,Binomial[2*k-2-n,k-2]]];c

%t (* sequence *)

%t 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 Row sums: A088218, alternating row sums: A091526.

%Y Central coefficients: binomial(3*n-2, n) (cf. A117671).

%Y T(n, 0) = binomial(2*(n-1), n) (cf. A001791).

%Y Cf. A257635.

%K nonn,tabl

%O 0,7

%A _Peter Luschny_, Nov 27 2021