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A209238 G.f.: A(x) = Sum_{n>=0} x^n * A(n*x) / A(x)^n.

%I A209238 #42 Jul 01 2025 16:11:13
%S A209238 1,1,1,1,2,6,33,261,3125,54281,1360227,49213679,2551724168,
%T A209238 189734190172,20202532257783,3077389342754387,670727373110373402,
%U A209238 209089605110085626174,93228754516671937710239,59465333534426105454885059,54262238803939167296498684986
%N A209238 G.f.: A(x) = Sum_{n>=0} x^n * A(n*x) / A(x)^n.
%e A209238 G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 33*x^6 + 261*x^7 + 3125*x^8 +...
%e A209238 The table of coefficients of x^k in A(n*x)/A(x)^n begin:
%e A209238   n=0: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A209238   n=1: [1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...];
%e A209238   n=2: [1, 0, 1, 2, 18, 128, 1704, 28908, 729461, 26121996, ...];
%e A209238   n=3: [1, 0, 3, 8, 102, 1032, 20052, 502104, 18852177, ...];
%e A209238   n=4: [1, 0, 6, 20, 333, 4404, 113292, 3769344, 188468286, ...];
%e A209238   n=5: [1, 0, 10, 40, 825, 13504, 433060, 17986560, 1123704270, ...];
%e A209238   n=6: [1, 0, 15, 70, 1725, 33684, 1294525, 64474290, ...];
%e A209238   n=7: [1, 0, 21, 112, 3213, 72912, 3266599, 189725472, ...];
%e A209238   n=8: [1, 0, 28, 168, 5502, 142296, 7282212, 483225336, ...];
%e A209238   n=9: [1, 0, 36, 240, 8838, 256608, 14768652, 1102246128,  ...]; ...
%e A209238 This sequence equals the antidiagonal sums of the above table.
%o A209238 (PARI) {a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=sum(k=0, n, x^k*subst(A,x,k*x +x*O(x^n))/A^(k))  );polcoeff(A, n)}
%o A209238 for(n=0, 30, print1(a(n), ", "))
%Y A209238 Cf. A125282.
%K A209238 nonn
%O A209238 0,5
%A A209238 _Paul D. Hanna_, Jan 27 2013