cp's OEIS Frontend

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).

%I A318301 #13 Sep 07 2018 04:24:39
%S A318301 1,1,1,2,3,5,10,18,33,61,122,234,450,867,1673,3346,6570,12906,25362,
%T A318301 49857,98041,196082,388818,771066,1529226,3033090,6016323,11934605,
%U A318301 23869210,47542338,94695858,188620650,275712074,748391058,1490765793,2969596981,5939193962,11854518714
%N 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).
%C A318301 The left edge of the triangle appears to be A005321.
%F A318301 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 A318301 Triangle begins:
%e A318301      1
%e A318301      1    1
%e A318301      2    3     5
%e A318301     10   18    33    61
%e A318301    122  234   450   867  1673
%e A318301   3346 6570 12906 25362 49857 98041
%e A318301   ...
%e A318301 T(5, 2) = (3346 + 6570) + (450 + 867 + 1673) = 12906;
%e A318301 T(5, 2) = 2 * 6570 - 234 = 12906.
%o A318301 (Python)
%o A318301 def T(n, k):
%o A318301     if k == 0:
%o A318301         if n == 0 or n == 1:
%o A318301             return 1
%o A318301         return 2 * T(n-1, n-1)
%o A318301     return 2 * T(n, k-1) - T(n-1, k-1)
%o A318301 (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 A318301 tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n,k), ", ")); print); \\ _Michel Marcus_, Aug 25 2018
%Y A318301 Cf. A005321.
%K A318301 nonn,tabl
%O A318301 0,4
%A A318301 _Nicolas Nagel_, Aug 24 2018