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 A182964 G.f.: x = Sum_{n>=1} a(n)*G_n(x)^n where G_n(x) is the n-th iteration of (x-x^2). 0

%I #7 Mar 12 2022 13:21:29

%S 1,1,4,28,278,3558,55600,1024459,21721922,520585084,13908591504,

%T 409749226342,13192559204352,460792929469356,17351608517036848,

%U 700669956538570048,30200986529461293238,1383912736505236230958

%N G.f.: x = Sum_{n>=1} a(n)*G_n(x)^n where G_n(x) is the n-th iteration of (x-x^2).

%C Compare g.f. of this sequence to the g.f. of A000272(n) = n^(n-2):

%C . x = Sum_{n>=1} n^(n-2)*[x/(1+n*x)]^n

%C where x/(1+n*x) = n-th iteration of x/(1+x).

%e G.f.: x = 1*G_1(x) + 1*G_2(x)^2 + 4*G_3(x)^3 + 28*G_4(x)^4 + 278*G_5(x)^5 + 3558*G_6(x)^6 + 55600*G_7(x)^7 +...+ a(n)*G_n(x)^n +...

%e where the initial iterations of G(x) = (x-x^2) begin:

%e G_1(x) = x - x^2;

%e G_2(x) = x - 2*x^2 + 2*x^3 - x^4;

%e G_3(x) = x - 3*x^2 + 6*x^3 - 9*x^4 + 10*x^5 - 8*x^6 + 4*x^7 - x^8;

%e G_4(x) = x - 4*x^2 + 12*x^3 - 30*x^4 + 64*x^5 - 118*x^6 +-...;

%e G_5(x) = x - 5*x^2 + 20*x^3 - 70*x^4 + 220*x^5 - 630*x^6 +-...;

%e ...

%e Coefficients in the n-th power of the n-th iteration of x-x^2 begin:

%e G_1(x)^1: [1, -1, 0, 0, 0, 0, 0, ...];

%e G_2(x)^2: [1, -4, 8, -10, 8, -4, 1, 0, 0, 0, 0, 0, ...];

%e G_3(x)^3: [1, -9, 45, -162, 462, -1095, 2217, -3900, 6024, ...];

%e G_4(x)^4: [1, -16, 144, -952, 5120, -23608, 96296, -354664, ...];

%e G_5(x)^5: [1, -25, 350, -3600, 30225, -218775, 1410780, ...];

%e G_6(x)^6: [1, -36, 720, -10530, 125400, -1286376, 11759727, ...];

%e ...

%o (PARI) /* n-th Iteration of a function: */

%o {ITERATE(n, F, p)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}

%o /* G.f.: */

%o {a(n)=local(G=x);if(n<=1,G=x,G=x-sum(k=1,n-1,a(k)*ITERATE(k,x-x^2,n)^k)); polcoeff(G, n)}

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jan 01 2011

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Last modified September 25 13:12 EDT 2023. Contains 365647 sequences. (Running on oeis4.)