cp's OEIS Frontend

A378289 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+k,r) * binomial(r,n-r)/(n+r+k) for k > 0.

%I A378289 #16 Nov 22 2024 08:30:21
%S A378289 1,1,0,1,1,0,1,2,3,0,1,3,7,10,0,1,4,12,26,38,0,1,5,18,49,105,154,0,1,
%T A378289 6,25,80,210,444,654,0,1,7,33,120,363,927,1944,2871,0,1,8,42,170,575,
%U A378289 1672,4191,8734,12925,0,1,9,52,231,858,2761,7810,19305,40040,59345,0
%N A378289 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+k,r) * binomial(r,n-r)/(n+r+k) for k > 0.
%F A378289 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(2/k) * (1 + x * A_k(x)^(1/k)) )^k for k > 0.
%F A378289 G.f. of column k: B(x)^k where B(x) is the g.f. of A001002.
%F A378289 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 A378289 Square array begins:
%e A378289   1,   1,    1,    1,    1,     1,     1, ...
%e A378289   0,   1,    2,    3,    4,     5,     6, ...
%e A378289   0,   3,    7,   12,   18,    25,    33, ...
%e A378289   0,  10,   26,   49,   80,   120,   170, ...
%e A378289   0,  38,  105,  210,  363,   575,   858, ...
%e A378289   0, 154,  444,  927, 1672,  2761,  4290, ...
%e A378289   0, 654, 1944, 4191, 7810, 13325, 21385, ...
%o A378289 (PARI) T(n, k, t=2, u=1) = 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 A378289 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378289 Columns k=0..3 give A000007, A001002, A052706(n+2), A052703(n+3).
%Y A378289 Cf. A009766, A026300, A378290, A378291, A378292.
%K A378289 nonn,tabl
%O A378289 0,8
%A A378289 _Seiichi Manyama_, Nov 21 2024