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A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - x*y - x^2*y*f(x, y+1)).

%I A357438 #17 Sep 29 2022 03:50:52
%S A357438 1,0,1,0,1,1,0,1,3,1,0,2,6,6,1,0,5,16,20,10,1,0,15,51,71,50,15,1,0,52,
%T A357438 186,281,231,105,21,1,0,203,759,1223,1114,616,196,28,1,0,877,3409,
%U A357438 5795,5701,3564,1428,336,36,1,0,4140,16655,29634,31011,21187,9780
%N A357438 Triangle T(n,k) read by rows, defined by the equation f(x, y) := Sum_{n, k} T(n, k) * y^k * x^n = 1/(1 - x*y - x^2*y*f(x, y+1)).
%C A357438 Row sums are A000110.
%F A357438 f(x, -1) = 1/(1 + x + x^2).
%F A357438 x + x^2*f(x, 2) = 1 - 1/f(x, 1) is g.f. for A074664.
%e A357438 Triangle starts:
%e A357438   1,
%e A357438   0,   1,
%e A357438   0,   1,    1,
%e A357438   0,   1,    3,    1,
%e A357438   0,   2,    6,    6,    1,
%e A357438   0,   5,   16,   20,   10,    1,
%e A357438   0,  15,   51,   71,   50,   15,    1,
%e A357438   0,  52,  186,  281,  231,  105,   21,  1,
%e A357438   0, 203,  759, 1223, 1114,  616,  196,  28,  1,
%e A357438   0, 877, 3409, 5795, 5701, 3564, 1428, 336, 36, 1,
%e A357438   ...
%t A357438 T[ n_, k_] := If[n < 0, 0, Coefficient[SeriesCoefficient[ Nest[ 1/(1 - x*y - x^2*y*(#/.y -> y+1))&, 1 + O[x], Ceiling[n/2]], {x, 0, n}], y, k]];
%o A357438 (PARI) {T(n, k) = if(n < 0, 0, f = 1 + O(x); forstep(i=1, n, 2, f = 1/(1 - x*y - x^2*y*subst(f, y, y+1))); polcoef(polcoef(f, n), k))};
%Y A357438 Cf. A000110, A049347, A074664.
%K A357438 nonn,tabl
%O A357438 1,9
%A A357438 _Michael Somos_, Sep 27 2022