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A378236 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(n+2*r+k,n)/(n+2*r+k) for k > 0.

%I A378236 #29 Nov 21 2024 11:15:24
%S A378236 1,1,0,1,2,0,1,4,8,0,1,6,20,44,0,1,8,36,120,280,0,1,10,56,236,800,
%T A378236 1936,0,1,12,80,400,1656,5696,14128,0,1,14,108,620,2960,12192,42416,
%U A378236 107088,0,1,16,140,904,4840,22592,92960,326304,834912,0,1,18,176,1260,7440,38352,176800,727824,2572992,6652608,0
%N A378236 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(n+2*r+k,n)/(n+2*r+k) for k > 0.
%F A378236 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(2/k)) )^k for k > 0.
%F A378236 G.f. of column k: B(x)^k where B(x) is the g.f. of A346626.
%F A378236 B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+2). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+2) for n > 0.
%e A378236 Square array begins:
%e A378236    1,     1,     1,     1,      1,      1,      1, ...
%e A378236    0,     2,     4,     6,      8,     10,     12, ...
%e A378236    0,     8,    20,    36,     56,     80,    108, ...
%e A378236    0,    44,   120,   236,    400,    620,    904, ...
%e A378236    0,   280,   800,  1656,   2960,   4840,   7440, ...
%e A378236    0,  1936,  5696, 12192,  22592,  38352,  61248, ...
%e A378236    0, 14128, 42416, 92960, 176800, 308560, 507152, ...
%o A378236 (PARI) T(n, k, t=1, u=2) = 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 A378236 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378236 Columns k=0..1 give A000007, A346626.
%Y A378236 Cf. A033877, A071949, A378237, A378238, A378239, A378240.
%K A378236 nonn,tabl
%O A378236 0,5
%A A378236 _Seiichi Manyama_, Nov 20 2024