login
Expansion of g.f. A(x) satisfying A(x - A(x)) = Sum_{n>=1} x^prime(n).
1

%I #8 Dec 30 2023 09:30:18

%S 1,3,14,97,801,7519,77362,854835,10008606,123019640,1576822761,

%T 20971869512,288325729119,4085343085563,59517261455725,

%U 889803062724848,13630085879852376,213643712913777884,3422884875842433236,56001632976304058623,934906290172540190490,15914504442059742698374

%N Expansion of g.f. A(x) satisfying A(x - A(x)) = Sum_{n>=1} x^prime(n).

%H Paul D. Hanna, <a href="/A367383/b367383.txt">Table of n, a(n) for n = 2..201</a>

%F G.f. A(x) = Sum_{n>=2} a(n)*x^n satisfies the following formulas.

%F (1) A(x - A(x)) = Sum_{n>=1} x^prime(n).

%F (2) A(x) = Sum_{n>=1} B(x)^prime(n) where B(x) = x + A(B(x)) and B(x - A(x)) = x.

%e G.f.: A(x) = x^2 + 3*x^3 + 14*x^4 + 97*x^5 + 801*x^6 + 7519*x^7 + 77362*x^8 + 854835*x^9 + 10008606*x^10 + 123019640*x^11 + ...

%e where

%e A(x - A(x)) = x^2 + x^3 + x^5 + x^7 + x^11 + x^13 + x^17 + ... + x^prime(n) + ...

%e also

%e A(x) = B(x)^2 + B(x)^3 + B(x)^5 + B(x)^7 + B(x)^11 + ... + B(x)^prime(n) + ...

%e where B(x) = x + A(B(x)) begins

%e B(x) = x + x^2 + 5*x^3 + 34*x^4 + 285*x^5 + 2712*x^6 + 28301*x^7 + 316987*x^8 + 3758998*x^9 + 46755855*x^10 + ...

%e Further,

%e B(B(x)) = x + 2*x^2 + 12*x^3 + 94*x^4 + 874*x^5 + 9061*x^6 + 101728*x^7 + 1214974*x^8 + 15260651*x^9 + 200003422*x^10 + ...

%e where

%e B(x) = x + B(B(x))^2 + B(B(x))^3 + B(B(x))^5 + B(B(x))^7 + ... + B(B(x))^prime(n) + ...

%o (PARI) \\ Generates N terms of this sequence \\

%o N = 30; A = x^2; B = x;

%o for(i=1,N, B = serreverse(x - A +x*O(x^N)); A = sum(n=1,N, B^prime(n)) ); Vec(A)

%K nonn

%O 2,2

%A _Paul D. Hanna_, Dec 29 2023