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A381569 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 A381570.

%I A381569 #12 Feb 28 2025 07:34:05
%S A381569 1,1,0,1,3,0,1,6,12,0,1,9,33,82,0,1,12,63,236,732,0,1,15,102,489,2100,
%T A381569 7944,0,1,18,150,868,4428,22248,99156,0,1,21,207,1400,8121,46422,
%U A381569 270268,1381464,0,1,24,273,2112,13665,85272,552540,3668568,21065853,0
%N A381569 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 A381570.
%F A381569 A(n,0) = 0^n; A(n,k) = k * Sum_{j=0..n} binomial(3*n-3*j+3*k,j)/(n-j+k) * A(n-j,j).
%e A381569 Square array begins:
%e A381569   1,    1,     1,     1,     1,      1, ...
%e A381569   0,    3,     6,     9,    12,     15, ...
%e A381569   0,   12,    33,    63,   102,    150, ...
%e A381569   0,   82,   236,   489,   868,   1400, ...
%e A381569   0,  732,  2100,  4428,  8121,  13665, ...
%e A381569   0, 7944, 22248, 46422, 85272, 145143, ...
%o A381569 (PARI) a(n, k) = if(k==0, 0^n, k*sum(j=0, n, binomial(3*n-3*j+3*k, j)/(n-j+k)*a(n-j, j)));
%Y A381569 Columns k=0..1 give A000007, A381570.
%Y A381569 Cf. A381566, A381567.
%K A381569 nonn,tabl
%O A381569 0,5
%A A381569 _Seiichi Manyama_, Feb 28 2025