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Irregular triangle read by rows: T(n,k) = A008949(n-1,k) if 0 <= k <= n - 2 otherwise A008949(n-1,2*n-4-k) if n - 2 <= k <= 2*n - 4.
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%I #6 Sep 21 2024 02:23:16

%S 1,1,3,1,1,4,7,4,1,1,5,11,15,11,5,1,1,6,16,26,31,26,16,6,1,1,7,22,42,

%T 57,63,57,42,22,7,1,1,8,29,64,99,120,127,120,99,64,29,8,1,1,9,37,93,

%U 163,219,247,255,247,219,163,93,37,9,1,1,10,46,130,256,382,466,502,511,502,466,382,256,130,46,10,1

%N Irregular triangle read by rows: T(n,k) = A008949(n-1,k) if 0 <= k <= n - 2 otherwise A008949(n-1,2*n-4-k) if n - 2 <= k <= 2*n - 4.

%H Nsibiet E. Udo, Praise Adeyemo, Balazs Szendroi, and Stavros Argyrios Papadakis, <a href="https://arxiv.org/abs/2409.10278">Ideals, representations and a symmetrised Bernoulli triangle</a>, arXiv:2409.10278 [math.AC], 2024. See p. 2.

%F Sum_{k=0..2*n-4} T(n,k) = A000337(n-1). [Udo et al.]

%e The triangle begins as:

%e 1;

%e 1, 3, 1;

%e 1, 4, 7, 4, 1;

%e 1, 5, 11, 15, 11, 5, 1;

%e 1, 6, 16, 26, 31, 26, 16, 6, 1;

%e 1, 7, 22, 42, 57, 63, 57, 42, 22, 7, 1;

%e ...

%t b[n_,k_]:=Sum[Binomial[n,j],{j,0,k}]; T[n_,k_]:=If[0<=k<=n-2,b[n-1,k],b[n-1,2n-4-k]]; Table[T[n,k],{n,2,10},{k,0,2n-4}]//Flatten

%Y Cf. A000337, A008949.

%K nonn,easy,tabf

%O 2,3

%A _Stefano Spezia_, Sep 20 2024