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A339350 Triangle read by rows. T(n,k) = Sum_{j=0..k} binomial(k-j+2, 2)*T(n-1, j), for n>=0, 0<=k<=n, with T(0,0)=1 and T(n,n)=0 for n>0.

%I A339350 #21 Feb 23 2024 08:34:24
%S A339350 1,1,0,1,3,0,1,6,15,0,1,9,39,91,0,1,12,72,272,612,0,1,15,114,570,1995,
%T A339350 4389,0,1,18,165,1012,4554,15180,32890,0,1,21,225,1625,8775,36855,
%U A339350 118755,254475,0,1,24,294,2436,15225,75516,302064,949344,2017356,0
%N A339350 Triangle read by rows. T(n,k) = Sum_{j=0..k} binomial(k-j+2, 2)*T(n-1, j), for n>=0, 0<=k<=n, with T(0,0)=1 and T(n,n)=0 for n>0.
%H A339350 Paolo Xausa, <a href="/A339350/b339350.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150 of the triangle, flattened).
%H A339350 Francesca Aicardi, <a href="https://arxiv.org/abs/2011.14628">Catalan triangle and tied arc diagrams</a>, arXiv:2011.14628 [math.CO], 2020.
%H A339350 Francesca Aicardi, <a href="https://arxiv.org/abs/2310.07317">Fuss-Catalan Triangles</a>, arXiv:2310.07317 [math.CO], 2023.
%e A339350 Triangle begins:
%e A339350   1;
%e A339350   1,  0;
%e A339350   1,  3,  0;
%e A339350   1,  6, 15,   0;
%e A339350   1,  9, 39,  91,   0;
%e A339350   1, 12, 72, 272, 612, 0;
%e A339350   ...
%t A339350 A339350[n_, k_] := A339350[n, k] = Which[k == 0, 1, n == k, 0, True, Sum[Binomial[k-j+2, 2]*A339350[n-1, j], {j, 0, k}]];
%t A339350 Table[A339350[n, k], {n, 0, 10}, {k, 0, n}] (* _Paolo Xausa_, Feb 23 2024 *)
%o A339350 (PARI) T(n,k) = if ((n==0) && (k==0), 1, if (n<=k, 0, sum(j=0, k, binomial(k-j+2, 2)*T(n-1, j))));
%Y A339350 Cf. subdiagonals: A006632, A006633, A006634, A006635.
%Y A339350 Cf. A002293 (row sums).
%K A339350 nonn,tabl
%O A339350 0,5
%A A339350 _Michel Marcus_, Dec 01 2020