A378321 - OEIS

%I #12 Nov 23 2024 09:27:38

%S 1,1,0,1,1,0,1,2,1,0,1,3,3,3,0,1,4,6,8,6,0,1,5,10,16,19,16,0,1,6,15,

%T 28,42,50,42,0,1,7,21,45,79,114,137,114,0,1,8,28,68,135,224,322,380,

%U 322,0,1,9,36,98,216,401,652,918,1088,918,0,1,10,45,136,329,672,1205,1912,2673,3152,2673,0

%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-3*r+k,r) * binomial(r,n-r)/(3*n-3*r+k) for k > 0.

%F G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x + x^2 * 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 A019497.

%F B(x)^k = B(x)^(k-1) + x * B(x)^(k-1) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k-1) + T(n-2,k+2) 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,  1,   3,   6,  10,   15,   21, ...

%e   0,  3,   8,  16,  28,   45,   68, ...

%e   0,  6,  19,  42,  79,  135,  216, ...

%e   0, 16,  50, 114, 224,  401,  672, ...

%e   0, 42, 137, 322, 652, 1205, 2088, ...

%o (PARI) T(n, k, t=0, 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, A019497.

%Y Cf. A071919, A378320.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Nov 23 2024