A378238 - OEIS

%I #33 Nov 28 2024 09:06:12

%S 1,1,0,1,2,0,1,4,14,0,1,6,32,134,0,1,8,54,324,1482,0,1,10,80,578,3696,

%T 17818,0,1,12,110,904,6810,45316,226214,0,1,14,144,1310,11008,85278,

%U 583152,2984206,0,1,16,182,1804,16490,140936,1113854,7769348,40503890,0

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

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

%F G.f. of column k: B(x)^k where B(x) is the g.f. of A144097.

%F B(x)^k = B(x)^(k-1) + x * B(x)^(k+2) + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k+2) + T(n-1,k+3) for n > 0.

%e Square array begins:

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

%e   0,      2,      4,       6,       8,      10,      12, ...

%e   0,     14,     32,      54,      80,     110,     144, ...

%e   0,    134,    324,     578,     904,    1310,    1804, ...

%e   0,   1482,   3696,    6810,   11008,   16490,   23472, ...

%e   0,  17818,  45316,   85278,  140936,  216002,  314700, ...

%e   0, 226214, 583152, 1113854, 1870352, 2914790, 4320608, ...

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

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

%Y Columns k=0..3 give A000007, A144097, A371675, A365843.

%Y T(n,n) gives 1/4 * A370102(n) for n > 0.

%Y Cf. A033877, A071949, A378236, A378237, A378239, A378240.

%K nonn,tabl,new

%O 0,5

%A _Seiichi Manyama_, Nov 20 2024