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EXAMPLE
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G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 143*x^4 + 1569*x^5 + 20911*x^6 +...
A(x) satisfies: A(G(x)) = G(x)/x where G(x) is the g.f. of A000699:
G(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 + 38232*x^7 +...
which in turn satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].
...
The table of coefficients in the initial powers of g.f. A(x) begins:
A^1: [1, 1, 3, 17, 143, 1569, 20911, 324673, 5720319, ...];
A^2: [1, 2, 7, 40, 329, 3526, 46107, 705444, 12289245, ...];
A^3: [1, 3, 12, 70, 567, 5949, 76350, 1150920, 19818801, ...];
A^4: [1, 4, 18, 108, 867, 8928, 112524, 1671000, 28436085, ...];
A^5: [1, 5, 25, 155, 1240, 12566, 155650, 2277050, 38285275, ...];
A^6: [1, 6, 33, 212, 1698, 16980, 206902, 2982096, 49529871, ...];
A^7: [1, 7, 42, 280, 2254, 22302, 267624, 3801036, 62355195, ...];
A^8: [1, 8, 52, 360, 2922, 28680, 339348, 4750872, 76971171, ...];
...
In the above table, notice that the main diagonal:
[1, 2, 12, 108, 1240, 16980, 267624, 4750872, 93615408, ...]
is related to the secondary diagonal in the following way:
[2*1, 4*3, 6*18, 8*155, 10*1698, 12*22302, 14*339348, 16*5850963, ...].
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PROG
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(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=(2*(#A-1)*Vec(Ser(A)^(#A))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A)); A[n+1]}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A*(A + x*A')/(A - x*A' +x*O(x^n))); polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
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