cp's OEIS Frontend

A384721 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A384719.

%I A384721 #13 Jun 08 2025 10:50:00
%S A384721 1,1,0,1,1,0,1,2,5,0,1,3,12,61,0,1,4,21,152,1281,0,1,5,32,279,3200,
%T A384721 39641,0,1,6,45,448,5937,98192,1655713,0,1,7,60,665,9696,181563,
%U A384721 4053688,88312869,0,1,8,77,936,14705,296864,7430265,213600200,5792082817,0
%N A384721 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A384719.
%F A384721 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} (n-j+k)^(j-1) * binomial(n,j) * A(n-j,2*j).
%e A384721 Square array begins:
%e A384721   1,     1,     1,      1,      1,      1, ...
%e A384721   0,     1,     2,      3,      4,      5, ...
%e A384721   0,     5,    12,     21,     32,     45, ...
%e A384721   0,    61,   152,    279,    448,    665, ...
%e A384721   0,  1281,  3200,   5937,   9696,  14705, ...
%e A384721   0, 39641, 98192, 181563, 296864, 452525, ...
%o A384721 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, (n-j+k)^(j-1)*binomial(n, j)*a(n-j, 2*j)));
%Y A384721 Columns k=0..1 give A000007, A384719.
%Y A384721 Cf. A380178, A384722.
%Y A384721 Cf. A384718, A384692.
%K A384721 nonn,tabl
%O A384721 0,8
%A A384721 _Seiichi Manyama_, Jun 08 2025