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Regular triangle read by rows, T(n,k) = T(n,k-1)+2*T(n-1,k)-T(n-1,k-1) for 1<=k<=n-2 with T(n,n)=T(n,n-1)=T(n,n-2) for n>=3 and T(1,1)=T(2,1)=T(2,2)=1.
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%I #49 Jun 02 2024 11:00:10

%S 1,1,1,2,2,2,4,6,6,6,8,16,22,22,22,16,40,68,90,90,90,32,96,192,304,

%T 394,394,394,64,224,512,928,1412,1806,1806,1806,128,512,1312,2656,

%U 4552,6752,8558,8558,8558,256,1152,3264,7264,13712,22664,33028,41586,41586,41586

%N Regular triangle read by rows, T(n,k) = T(n,k-1)+2*T(n-1,k)-T(n-1,k-1) for 1<=k<=n-2 with T(n,n)=T(n,n-1)=T(n,n-2) for n>=3 and T(1,1)=T(2,1)=T(2,2)=1.

%H Dongsu Kim and Zhicong Lin, <a href="https://arxiv.org/abs/1706.07208">Refined restricted inversion sequences</a>, arXiv:1706.07208 [math.CO], 2017-2020.

%H Toufik Mansour and Mark Shattuck, <a href="https://arxiv.org/abs/2104.04491">On a conjecture of Lin and Kim concerning a refinement of Schröder numbers</a>, arXiv:2104.04491 [math.CO], 2021.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 2, 2;

%e 4, 6, 6, 6;

%e 8, 16, 22, 22, 22;

%e 16, 40, 68, 90, 90, 90;

%e ...

%o (PARI)

%o T(n,k) = if (k>=1, if (n<=2, 1, if (k<=n-2, T(n,k-1)+2*T(n-1,k)-T(n-1,k-1), T(n,n-2))));

%o tabl(nn) = {for (n=1, nn, for (k=1, n, print1(T(n,k), ", ");); print;);}

%Y Cf. A011782 (1st column), A155069 (right diagonal).

%K nonn,tabl,easy

%O 1,4

%A _Michel Marcus_, Apr 12 2021