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%I #13 Sep 07 2018 04:24:39
%S 1,1,1,2,3,5,10,18,33,61,122,234,450,867,1673,3346,6570,12906,25362,
%T 49857,98041,196082,388818,771066,1529226,3033090,6016323,11934605,
%U 23869210,47542338,94695858,188620650,275712074,748391058,1490765793,2969596981,5939193962,11854518714
%N Triangle T(n, k) read by rows: T(0, 0) = 1 and T(n, k) = Sum_{i=0..k-1} T(n, i) + Sum_{i=k..n-1} T(n-1, i).
%C The left edge of the triangle appears to be A005321.
%F An equivalent recursion: T(0, 0) = T(1, 0) = 1, T(n, 0) = 2*T(n-1, n-1) if n>=2, T(n, k) = 2*T(n, k-1) - T(n-1, k-1) if n>=k>=1.
%e Triangle begins:
%e 1
%e 1 1
%e 2 3 5
%e 10 18 33 61
%e 122 234 450 867 1673
%e 3346 6570 12906 25362 49857 98041
%e ...
%e T(5, 2) = (3346 + 6570) + (450 + 867 + 1673) = 12906;
%e T(5, 2) = 2 * 6570 - 234 = 12906.
%o (Python)
%o def T(n, k):
%o if k == 0:
%o if n == 0 or n == 1:
%o return 1
%o return 2 * T(n-1, n-1)
%o return 2 * T(n, k-1) - T(n-1, k-1)
%o (PARI) T(n, k) = if (k == 0, if (n <= 1, 1, 2 * T(n-1, n-1)), 2 * T(n, k-1) - T(n-1, k-1));
%o tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Aug 25 2018
%Y Cf. A005321.
%K nonn,tabl
%O 0,4
%A _Nicolas Nagel_, Aug 24 2018
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