A378291 - OEIS

%I #11 Nov 22 2024 08:31:12

%S 1,1,0,1,1,0,1,2,2,0,1,3,5,6,0,1,4,9,16,20,0,1,5,14,31,56,72,0,1,6,20,

%T 52,114,208,273,0,1,7,27,80,201,438,806,1073,0,1,8,35,116,325,800,

%U 1739,3220,4333,0,1,9,44,161,495,1341,3260,7077,13168,17869,0

%N Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,0) = 0^n and T(n,k) = k * Sum_{r=0..n} binomial(3*n-2*r+k,r) * binomial(r,n-r)/(3*n-2*r+k) for k > 0.

%F G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + x * A_k(x)^(3/k)) )^k for k > 0.

%F G.f. of column k: B(x)^k where B(x) is the g.f. of A186996.

%F B(x)^k = B(x)^(k-1) + x * B(x)^k + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-2,k+3) for n > 1.

%e Square array begins:

%e   1,   1,   1,    1,    1,    1,    1, ...

%e   0,   1,   2,    3,    4,    5,    6, ...

%e   0,   2,   5,    9,   14,   20,   27, ...

%e   0,   6,  16,   31,   52,   80,  116, ...

%e   0,  20,  56,  114,  201,  325,  495, ...

%e   0,  72, 208,  438,  800, 1341, 2118, ...

%e   0, 273, 806, 1739, 3260, 5615, 9119, ...

%o (PARI) T(n, k, t=1, u=3) = if(k==0, 0^n, k*sum(r=0, n, binomial(t*r+u*(n-r)+k, r)*binomial(r, n-r)/(t*r+u*(n-r)+k)));

%o matrix(7, 7, n, k, T(n-1, k-1))

%Y Columns k=0..1 give A000007, A186996.

%Y Cf. A009766, A026300, A378289, A378290, A378292.

%Y Cf. A378237.

%K nonn,tabl,new

%O 0,8

%A _Seiichi Manyama_, Nov 21 2024