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%I #15 Oct 04 2023 04:49:05
%S 1,1,3,1,5,7,1,7,16,15,1,9,29,42,31,1,11,46,93,99,63,1,13,67,176,256,
%T 219,127,1,15,92,299,562,638,466,255,1,17,121,470,1093,1586,1486,968,
%U 511,1,19,154,697,1941,3473,4096,3302,1981,1023
%N T(n, k) = binomial(2*n + 1 - k, k)*hypergeom([1, 1, -k], [1, 2*(n - k + 1)], -1), triangle read by rows, T(n, k) for n >= 0 and 0 <= k <= n.
%e Triangle starts:
%e [0] 1
%e [1] 1, 3
%e [2] 1, 5, 7
%e [3] 1, 7, 16, 15
%e [4] 1, 9, 29, 42, 31
%e [5] 1, 11, 46, 93, 99, 63
%e [6] 1, 13, 67, 176, 256, 219, 127
%e [7] 1, 15, 92, 299, 562, 638, 466, 255
%e [8] 1, 17, 121, 470, 1093, 1586, 1486, 968, 511
%p T := (n, k) -> binomial(2*n + 1 - k, k)*hypergeom([1, 1, -k], [1, 2*(n-k+1)], -1):
%p for n from 0 to 11 do seq(simplify(T(n, k)), k = 0..n) od;
%t s={};For[n=0,n<19,n++,For[k=0,k<n+1,k++,AppendTo[s,2^(2*n+1-k)-Sum[Binomial[2*n+1-k,i],{i,0,2*n+1-2*k-1}]]]];s (* _Detlef Meya_, Oct 03 2023 *)
%Y Row sums are A105693(n-1).
%Y Cf. A097750.
%K nonn,tabl
%O 0,3
%A _Peter Luschny_, Oct 28 2018
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