A391318 - OEIS

%I #24 Dec 31 2025 16:11:41

%S 0,1,1,2,3,2,3,7,7,3,4,13,18,13,4,5,21,39,39,21,5,6,31,74,97,74,31,6,

%T 7,43,127,211,211,127,43,7,8,57,202,413,520,413,202,57,8,9,73,303,743,

%U 1145,1145,743,303,73,9,10,91,434,1249,2302,2811,2302,1249,434,91,10

%N Triangle array read by rows: T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 1 with T(0,0) = T(n,-1) = T(n,n+1) = 0.

%C Column k=1 appears to be A002061(n).

%C Column k=2 appears to be A051743(n-1) - A000332(n-2), starting from n=2.

%H Leonidas Liponis, <a href="/A391318/b391318.txt">Table of n, a(n) for n = 0..5150</a> (100 rows flattened)

%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-1) + 1 with T(0,0) = T(n,-1) = T(n,n-k) = 0.

%e Triangle begins:

%e   Sum | Terms

%e     0 |   0;

%e     2 |   1,   1;

%e     7 |   2,   3,   2;

%e    20 |   3,   7,   7,   3;

%e    52 |   4,  13,  18,  13,   4;

%e   130 |   5,  21,  39,  39,  21,   5;

%e   319 |   6,  31,  74,  97,  74,  31,   6;

%e   776 |   7,  43, 127, 211, 211, 127,  43,  7;

%e   ... |   ...

%t T[n_, k_] := T[n, k] = Piecewise[{{0, (n == 0 && k == 0) || k == -1 || n - k == -1}, {T[n - 1, k] + T[n - 1, k - 1] + T[n - 2, k - 1] + 1, n > 0 && k >= 0 && n - k >= 0}}];

%t TableForm[Table[T[n, k], {n, 0, 10}, {k, 0, n}], TableAlignments -> Center, TableSpacing -> {1, 1}]

%Y Cf. A391272 (row sums).

%K nonn,easy,tabl

%O 0,4

%A _Leonidas Liponis_, Dec 06 2025