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A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 + x)^k)^(1/3).

%I A361839 #12 Mar 27 2023 10:14:42
%S A361839 1,1,3,1,3,18,1,3,21,126,1,3,24,162,945,1,3,27,201,1341,7371,1,3,30,
%T A361839 243,1809,11529,58968,1,3,33,288,2352,16893,101619,480168,1,3,36,336,
%U A361839 2973,23607,161676,911466,3961386,1,3,39,387,3675,31818,242757,1574289,8281737,33011550
%N A361839 Square array T(n,k), n>=0, k>=0, read by antidiagonals downwards, where column k is the expansion of 1/(1 - 9*x*(1 + x)^k)^(1/3).
%F A361839 n*T(n,k) = 3 * Sum_{j=0..k} binomial(k,j)*(3*n-2-2*j)*T(n-1-j,k) for n > k.
%F A361839 T(n,k) = Sum_{j=0..n} (-9)^j * binomial(-1/3,j) * binomial(k*j,n-j).
%e A361839 Square array begins:
%e A361839      1,     1,     1,     1,     1,     1, ...
%e A361839      3,     3,     3,     3,     3,     3, ...
%e A361839     18,    21,    24,    27,    30,    33, ...
%e A361839    126,   162,   201,   243,   288,   336, ...
%e A361839    945,  1341,  1809,  2352,  2973,  3675, ...
%e A361839   7371, 11529, 16893, 23607, 31818, 41676, ...
%o A361839 (PARI) T(n,k) = sum(j=0, n, (-9)^j*binomial(-1/3, j)*binomial(k*j, n-j));
%Y A361839 Columns k=0..3 give A004987, A180400, A361841, A361842.
%Y A361839 Main diagonal gives A361846.
%Y A361839 Cf. A361830, A361840.
%K A361839 nonn,tabl
%O A361839 0,3
%A A361839 _Seiichi Manyama_, Mar 26 2023