A378292 - OEIS

%I #13 Nov 22 2024 08:31:06

%S 1,1,0,1,1,0,1,2,3,0,1,3,7,11,0,1,4,12,28,46,0,1,5,18,52,123,207,0,1,

%T 6,25,84,240,572,979,0,1,7,33,125,407,1155,2769,4797,0,1,8,42,176,635,

%U 2028,5733,13806,24138,0,1,9,52,238,936,3276,10332,29136,70414,123998,0

%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(2*n+k,r) * binomial(r,n-r)/(2*n+k) for k > 0.

%F G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(2/k)) )^k for k > 0.

%F G.f. of column k: B(x)^k where B(x) is the g.f. of A006605.

%F B(x)^k = B(x)^(k-1) + x * B(x)^(k+1) + x^2 * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+1) + 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,   3,    7,   12,    18,    25,    33, ...

%e   0,  11,   28,   52,    84,   125,   176, ...

%e   0,  46,  123,  240,   407,   635,   936, ...

%e   0, 207,  572, 1155,  2028,  3276,  4998, ...

%e   0, 979, 2769, 5733, 10332, 17140, 26860, ...

%o (PARI) T(n, k, t=2, u=2) = 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..2 give A000007, A006605, A143927.

%Y Cf. A009766, A026300, A378289, A378290, A378291.

%K nonn,tabl

%O 0,8

%A _Seiichi Manyama_, Nov 21 2024