cp's OEIS Frontend

A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.

%I A165997 #10 Mar 29 2023 08:58:29
%S A165997 1,1,1,1,1,2,2,1,3,5,1,4,9,9,1,5,14,23,23,1,6,20,43,66,1,7,27,70,136,
%T A165997 136,1,8,35,105,241,377,377,1,9,44,149,390,767,1144,1,10,54,203,593,
%U A165997 1360,2504,2504,1,11,65,268,861,2221,4725,7229,7229,1,12,77,345,1206
%N A165997 Irregular triangle read by rows: T(0,0) = 1, T(n,k) = T(n,k-1) + T(n-1,k) for n > 0, 0 < k <= f(n), where f(n) = floor((2*n+3)/3), and entries outside triangle are 0.
%C A165997 There are f(n) = floor((2*n+3)/3) = A004396(n+1) terms in row n.
%e A165997 Triangle begins:
%e A165997        k=0   1   2    3     4     5     6      7      8
%e A165997   n=0:   1
%e A165997   n=1:   1
%e A165997   n=2:   1,  1
%e A165997   n=3:   1,  2,  2
%e A165997   n=4:   1,  3,  5
%e A165997   n=5:   1,  4,  9,   9
%e A165997   n=6:   1,  5, 14,  23,   23
%e A165997   n=7:   1,  6, 20,  43,   66
%e A165997   n=8:   1,  7, 27,  70,  136,  136
%e A165997   n=9:   1,  8, 35, 105,  241,  377,  377
%e A165997   n=10:  1,  9, 44, 149,  390,  767, 1144
%e A165997   n=11:  1, 10, 54, 203,  593, 1360, 2504,  2504
%e A165997   n=12:  1, 11, 65, 268,  861, 2221, 4725,  7229,  7229
%e A165997   n=13:  1, 12, 77, 345, 1206, 3427, 8152, 15381, 22610
%e A165997   ...
%o A165997 (PARI) f(n) = floor((2*(n-1)+3)/3); s=14; M=matrix(s,s); for(n=1,s,M[n,1]=1); for(n=2,s,for(k=2,f(n),M[n,k]=M[n,k-1]+M[n-1,k])); for(n=1,s,for(k=1,f(n),print1(M[n,k],", ")))
%Y A165997 Cf. A004396 (row lengths).
%Y A165997 Cf. A060941 (diagonal T(3*n, 2*n)).
%K A165997 nonn,tabf
%O A165997 0,6
%A A165997 _Gerald McGarvey_, Oct 03 2009