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A375659 For 0<=k<=n, T(n,k) = the number of Dyck-type lattice paths of length n, starting at the point (0,k), triangle T read by rows.

%I A375659 #21 Sep 09 2024 19:58:18
%S A375659 1,1,2,2,3,4,3,6,7,8,6,10,14,15,16,10,20,25,30,31,32,20,35,50,56,62,
%T A375659 63,64,35,70,91,112,119,126,127,128,70,126,182,210,238,246,254,255,
%U A375659 256,126,252,336,420,456,492,501,510,511,512,252,462,672,792,912,957,1002,1012,1022,1023,1024
%N A375659 For 0<=k<=n, T(n,k) = the number of Dyck-type lattice paths of length n, starting at the point (0,k), triangle T read by rows.
%C A375659 A Dyck type lattice path has the steps (1,1) or (1,-1) and never passes below the x-axis.
%C A375659 For k>=n, the number of Dyck-type lattice paths is 2^n.
%C A375659 The sequence completes A322291 by adding a diagonal of powers of 2.
%H A375659 John Tyler Rascoe, <a href="/A375659/b375659.txt">Rows n = 0..140, flattened</a>
%F A375659 T(n,k) = Sum_{i = 0..k} binomial(n, floor((n-k)/2)+i).
%F A375659 T(n,k) = T(n-1,k-1)+T(n-1,k+1), for all n>=2 and 1<=k<=n-2.
%e A375659   n | k=0    1    2    3    4    5    6    7
%e A375659  ---+---------------------------------------
%e A375659   0 |  1
%e A375659   1 |  1    2
%e A375659   2 |  2    3    4
%e A375659   3 |  3    6    7    8
%e A375659   4 |  6   10   14   15   16
%e A375659   5 | 10   20   25   30   31   32
%e A375659   6 | 20   35   50   56   62   63   64
%e A375659   7 | 35   70   92  112  119  126  127  128
%p A375659 a:=(n,k)->sum(binomial(n, floor((1/2)*(n-k))+i), i = 0..k):
%p A375659 seq(seq(a(n, k), k = 0..n), n = 0..11);
%o A375659 (Python)
%o A375659 from math import comb
%o A375659 def A375659(n,k):
%o A375659     return sum(comb(n,i+(n-k)//2) for i in range(k+1)) # _John Tyler Rascoe_, Sep 04 2024
%Y A375659 T(n,0) = T(n-1,1) = A001405(n).
%Y A375659 T(n,n) = A000079(n).
%Y A375659 T(n,n-1) = A000225(n).
%Y A375659 T(n,n-2) = A000918(n).
%Y A375659 T(n,n-3) = A000247(n).
%Y A375659 T(n,n-4) = A052515(n).
%Y A375659 Row sums = A189162(n+1).
%Y A375659 Cf. A322291, A368175.
%K A375659 nonn,tabl,easy
%O A375659 0,3
%A A375659 _Marilena Jianu_, Aug 23 2024