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A384777 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A382450.

%I A384777 #27 Jun 10 2025 13:57:14
%S A384777 1,1,0,1,1,0,1,2,3,0,1,3,7,19,0,1,4,12,44,221,0,1,5,18,76,489,4597,0,
%T A384777 1,6,25,116,813,9750,174007,0,1,7,33,165,1203,15543,358895,12328367,0,
%U A384777 1,8,42,224,1670,22072,555696,25040728,1674839513,0,1,9,52,294,2226,29446,765572,38156448,3375603329,443624694633,0
%N A384777 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of B(x)^k, where B(x) is the g.f. of A382450.
%F A384777 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} 2^(n-j) * binomial(n+k,j)/(n+k) * A(n-j,j).
%e A384777 Square array begins:
%e A384777   1,    1,    1,     1,     1,     1, ...
%e A384777   0,    1,    2,     3,     4,     5, ...
%e A384777   0,    3,    7,    12,    18,    25, ...
%e A384777   0,   19,   44,    76,   116,   165, ...
%e A384777   0,  221,  489,   813,  1203,  1670, ...
%e A384777   0, 4597, 9750, 15543, 22072, 29446, ...
%o A384777 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, 2^(n-j)*binomial(n+k,j)/(n+k)*a(n-j, j)));
%Y A384777 Columns k=0..1 give A000007, A382450.
%Y A384777 Cf. A379598.
%K A384777 nonn,tabl
%O A384777 0,8
%A A384777 _Seiichi Manyama_, Jun 10 2025