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G.f. A(x) satisfies: 1+x = A(x)^2 + A(x)^6 - A(x)^7.
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%I #11 Mar 23 2023 03:45:58

%S 1,1,5,65,1020,17915,336881,6634391,135085305,2820748975,60074608660,

%T 1299896748546,28496343865808,631545920570632,14126673078922850,

%U 318514279669755470,7231331734648562366,165172699475903928921,3793005825337785580290,87518313783310367308770

%N G.f. A(x) satisfies: 1+x = A(x)^2 + A(x)^6 - A(x)^7.

%H Robert Israel, <a href="/A249930/b249930.txt">Table of n, a(n) for n = 0..700</a>

%F G.f.: 1 + Series_Reversion(x - 5*x^2 - 15*x^3 - 20*x^4 - 15*x^5 - 6*x^6 - x^7).

%e G.f.: A(x) = 1 + x + 5*x^2 + 65*x^3 + 1020*x^4 + 17915*x^5 + ...

%e Related expansions.

%e A(x)^2 = 1 + 2*x + 11*x^2 + 140*x^3 + 2195*x^4 + 38520*x^5 + ...

%e A(x)^6 = 1 + 6*x + 45*x^2 + 560*x^3 + 8760*x^4 + 153546*x^5 + ...

%e A(x)^7 = 1 + 7*x + 56*x^2 + 700*x^3 + 10955*x^4 + 192066*x^5 + ...

%e where 1+x = A(x)^2 + A(x)^6 - A(x)^7.

%p S:= series(RootOf(1+x-(y^2+y^6-y^7),y,1),x,26):

%p seq(coeff(S,x,i),i=0..25); # _Robert Israel_, Mar 22 2023

%o (PARI) /* From 1+x = A(x)^2 + A(x)^6 - A(x)^7: */

%o {a(n)=local(A=[1, 1]); for(i=1, n, A=concat(A, 0); A[#A]=-Vec(Ser(A)^2+Ser(A)^6-Ser(A)^7)[#A]); A[n+1]}

%o for(n=0, 25, print1(a(n) , ", "))

%o (PARI) /* From Series Reversion: */

%o {a(n)=local(A=1+serreverse(x - 5*x^2 - 15*x^3 - 20*x^4 - 15*x^5 - 6*x^6 - x^7 + x^2*O(x^n)));polcoeff(A,n)}

%o for(n=0, 25, print1(a(n) , ", "))

%Y Cf. A249926, A249927, A249928, A249929, A249931, A249932.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 27 2014