cp's OEIS Frontend

A378323 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(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.

%I A378323 #11 Nov 23 2024 09:27:42
%S A378323 1,1,0,1,1,0,1,2,4,0,1,3,9,18,0,1,4,15,44,94,0,1,5,22,79,240,529,0,1,
%T A378323 6,30,124,450,1390,3135,0,1,7,39,180,737,2685,8404,19270,0,1,8,49,248,
%U A378323 1115,4532,16585,52426,121732,0,1,9,60,329,1599,7066,28624,105147,334964,785496,0
%N A378323 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(3r+k,r) * binomial(r,n-r)/(3*r+k) for k > 0.
%F A378323 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * (1 + x) * A_k(x)^(3/k) )^k for k > 0.
%F A378323 G.f. of column k: B(x)^k where B(x) is the g.f. of A364475.
%F A378323 B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x^2 * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-2,k+2) for n > 1.
%e A378323 Square array begins:
%e A378323   1,    1,    1,     1,     1,     1,     1, ...
%e A378323   0,    1,    2,     3,     4,     5,     6, ...
%e A378323   0,    4,    9,    15,    22,    30,    39, ...
%e A378323   0,   18,   44,    79,   124,   180,   248, ...
%e A378323   0,   94,  240,   450,   737,  1115,  1599, ...
%e A378323   0,  529, 1390,  2685,  4532,  7066, 10440, ...
%e A378323   0, 3135, 8404, 16585, 28624, 45655, 69021, ...
%o A378323 (PARI) T(n, k, t=3, u=0) = 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 A378323 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378323 Columns k=0..2 give A000007, A364475, A371576.
%Y A378323 Cf. A038137, A071943.
%Y A378323 Cf. A378318.
%K A378323 nonn,tabl
%O A378323 0,8
%A A378323 _Seiichi Manyama_, Nov 23 2024