cp's OEIS Frontend

A378237 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+3*r+k,n)/(n+3*r+k) for k > 0.

%I A378237 #29 Nov 21 2024 11:14:55
%S A378237 1,1,0,1,2,0,1,4,10,0,1,6,24,74,0,1,8,42,188,642,0,1,10,64,350,1680,
%T A378237 6082,0,1,12,90,568,3234,16212,60970,0,1,14,120,850,5440,31878,164584,
%U A378237 635818,0,1,16,154,1204,8450,54888,328426,1732172,6826690,0,1,18,192,1638,12432,87402,574848,3494142,18728352,74958914,0
%N A378237 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+3*r+k,n)/(n+3*r+k) for k > 0.
%F A378237 G.f. A_k(x) of column k satisfies A_k(x) = ( 1 + x * A_k(x)^(1/k) * (1 + A_k(x)^(3/k)) )^k for k > 0.
%F A378237 G.f. of column k: B(x)^k where B(x) is the g.f. of A349310.
%F A378237 B(x)^k = B(x)^(k-1) + x * B(x)^k + x * B(x)^(k+3). So T(n,k) = T(n,k-1) + T(n-1,k) + T(n-1,k+3) for n > 0.
%e A378237 Square array begins:
%e A378237    1,     1,      1,      1,      1,      1,       1, ...
%e A378237    0,     2,      4,      6,      8,     10,      12, ...
%e A378237    0,    10,     24,     42,     64,     90,     120, ...
%e A378237    0,    74,    188,    350,    568,    850,    1204, ...
%e A378237    0,   642,   1680,   3234,   5440,   8450,   12432, ...
%e A378237    0,  6082,  16212,  31878,  54888,  87402,  131964, ...
%e A378237    0, 60970, 164584, 328426, 574848, 931770, 1433544, ...
%o A378237 (PARI) T(n, k, t=1, u=3) = 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 A378237 matrix(7, 7, n, k, T(n-1, k-1))
%Y A378237 Columns k=0..1 give A000007, A349310.
%Y A378237 Cf. A033877, A071949, A378236, A378238, A378239, A378240.
%K A378237 nonn,tabl
%O A378237 0,5
%A A378237 _Seiichi Manyama_, Nov 20 2024