A288267 - OEIS

%I #32 Nov 14 2020 07:58:08

%S 1,1,1,1,1,2,2,1,3,3,1,5,5,2,9,9,4,1,15,15,6,1,26,26,11,2,45,45,19,4,

%T 78,78,33,7,1,135,135,57,12,1,234,234,99,21,2,406,406,172,37,4,704,

%U 704,298,64,7,1222,1222,518,112,13,1,2120,2120,898,194,22,1,3679

%N Triangle read by rows: T(n,k) = T(n,k+1) + T(n-k,k-1) with T(0,0) = 1 and T(n,k) = 0 if k<0 or k > A003056(n).

%H Seiichi Manyama, <a href="/A288267/b288267.txt">Rows n = 0..481, flattened</a>

%e First few rows are:

%e    1;

%e    1,  1;

%e    1,  1;

%e    2,  2, 1;

%e    3,  3, 1;

%e    5,  5, 2;

%e    9,  9, 4, 1;

%e   15, 15, 6, 1.

%p T:= proc(n,k) option remember; `if`(k<0 or k*(k+1)/2>n, 0,

%p       `if`(n=0, 1, T(n, k+1)+T(n-k, k-1)))

%p     end:

%p seq(seq(T(n, k), k=0..floor((sqrt(1+8*n)-1)/2)), n=0..20); # _Alois P. Heinz_, Sep 01 2017

%t T[n_, k_] := T[n, k] = If[k < 0 || k(k+1)/2 > n, 0, If[n == 0, 1, T[n, k+1] + T[n-k, k-1]]];

%t Table[T[n, k], {n, 0, 20}, {k, 0, Floor[(Sqrt[8n+1]-1)/2]}] // Flatten (* _Jean-François Alcover_, Nov 14 2020, after _Alois P. Heinz_ *)

%Y Columns 0+1,2 give A005169, A289080 (for n>0).

%Y Cf. A003056.

%K nonn,tabf

%O 0,6

%A _Seiichi Manyama_, Sep 01 2017