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A138054 Levels of substitution A059832 taken as polynomials p(x,n)]and coefficients of the differential polynomials returned as q(x,n) = dp(x,n)dx coefficients (first three zeros omitted).

%I A138054 #6 Sep 09 2023 19:35:57
%S A138054 2,6,3,2,6,12,1,4,9,8,15,6,14,24,2,6,6,12,5,12,21,24,9,20,33,24,39,14,
%T A138054 30,48,3,2,6,12,15,6,14,24,18,30,11,24,39,14,30,48,34,54,19,40,63,66,
%U A138054 23,48,75,52,81,28,58,90,1,4,9,8,15,6,14,24,9,20,33,24,39,14,30,48,51,18
%N A138054 Levels of substitution A059832 taken as polynomials p(x,n)]and coefficients of the differential polynomials returned as q(x,n) = dp(x,n)dx coefficients (first three zeros omitted).
%C A138054 Row sums are (with zeros) {0, 0, 0, 8, 23, 81, 305, 1027, 3514, 12002, 40658, ...}.
%C A138054 This sequence is as a result of my Pc Mandelbrot-Julia work.
%C A138054 I noticed that these substitution levels increased like iteration polynomials, so I converted the substitution levels to polynomials.
%C A138054 To get a good implicit plot I have been using the inverse of the differential in polynomials as a product.
%C A138054 So I used that kind of procedure to get the differentiation of a substitution.
%F A138054 p(x,n)=Sum[A059832[n,m]*t(m-1),{m,1,n}]; q(x,n)=dp(x,n)dx; out_n,m=Coefficients(q(x,n).
%e A138054 Three zeros then:
%e A138054 {2, 6},
%e A138054 {3, 2, 6, 12},
%e A138054 {1, 4, 9, 8, 15, 6, 14, 24},
%e A138054 {2, 6, 6, 12, 5, 12, 21, 24, 9, 20, 33, 24, 39, 14, 30, 48},
%t A138054 Clear[a, s, p, t, m, n] (* substitution *) s[1] = {2}; s[2] = {3}; s[3] = {1, 2, 3}; t[a_] := Flatten[s /@ a]; p[0] = {1}; p[1] = t[p[0]]; p[n_] := t[p[n - 1]]; (*A059832*) a = Table[p[n], {n, 0, 10}]; Flatten[a]; b = Table[CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]*x^( m - 1), {m, 1, Length[a[[n]]]}]], x], x], {n, 1, 11}]; Flatten[b] Table[Apply[Plus, CoefficientList[D[Apply[Plus, Table[a[[n]][[m]]* x^(m - 1), {m, 1, Length[a[[n]]]}]], x], x]], {n, 1, 11}];
%Y A138054 Cf. A059832, A105256.
%K A138054 nonn,uned,tabf
%O A138054 1,1
%A A138054 _Roger L. Bagula_ and _Gary W. Adamson_, May 02 2008