A318301 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).
1, 1, 1, 2, 3, 5, 10, 18, 33, 61, 122, 234, 450, 867, 1673, 3346, 6570, 12906, 25362, 49857, 98041, 196082, 388818, 771066, 1529226, 3033090, 6016323, 11934605, 23869210, 47542338, 94695858, 188620650, 275712074, 748391058, 1490765793, 2969596981, 5939193962, 11854518714
Offset: 0
Examples
Triangle begins:
1
1 1
2 3 5
10 18 33 61
122 234 450 867 1673
3346 6570 12906 25362 49857 98041
...
T(5, 2) = (3346 + 6570) + (450 + 867 + 1673) = 12906;
T(5, 2) = 2 * 6570 - 234 = 12906.
Crossrefs
Cf. A005321.
Programs
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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)); tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ Michel Marcus, Aug 25 2018
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Python
def T(n, k): if k == 0: if n == 0 or n == 1: return 1 return 2 * T(n-1, n-1) return 2 * T(n, k-1) - T(n-1, k-1)
Formula
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.
Comments