A375456 - OEIS

%I #15 Oct 13 2024 13:25:37

%S 1,1,5,40,414,5096,71465,1113432,18964415,349252420,6899717360,

%T 145360352592,3250782038728,76887080836140,1917401350590001,

%U 50284361717695424,1383636099826635216,39865319955874291412,1200467734347938040895,37718141663144558046536,1234556743772762830508484

%N Expansion of g.f. A(x) satisfying x = A( A(x) - 2*A(x)^2*A'(x) ).

%C It appears that a(A219608(n)) is odd for n >= 1, and that the only other odd term is a(2) = 1.

%H Paul D. Hanna, <a href="/A375456/b375456.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) x = A( A(x) - 2*A(x)^2 * A'(x) ).

%F (2) A(A(x)) = x + 2*A(A(x))^2 * A'(A(x)).

%F (3) R(R(x)) = x - 2*x^2 * A'(R(x)), where A(R(x)) = x.

%F a(n) ~ c * d^n * n! * n^alpha, where d = 1.3534821142256737694364485294..., alpha = 2.7625501039589..., c = 0.0101323266748276... - _Vaclav Kotesovec_, Sep 07 2024

%e G.f.: A(x) = x + x^2 + 5*x^3 + 40*x^4 + 414*x^5 + 5096*x^6 + 71465*x^7 + 1113432*x^8 + 18964415*x^9 + 349252420*x^10 + ...

%e where x = A( A(x) - 2*A(x)^2 * A'(x) ).

%e RELATED SERIES.

%e Let R(x) be the series reversion of A(x) so that R(A(x)) = x, then

%e R(x) = x - x^2 - 3*x^3 - 20*x^4 - 190*x^5 - 2240*x^6 - 30759*x^7 - 475116*x^8 - 8081145*x^9 - 149243380*x^10 + ...

%e where R(x) = A(x) - 2*A(x)^2 * A'(x).

%e A(x)^2 = x^2 + 2*x^3 + 11*x^4 + 90*x^5 + 933*x^6 + 11420*x^7 + 158862*x^8 + 2453874*x^9 + ...

%o (PARI) {a(n) = my(A=[0,1],Ax=x); for(i=1,n, A=concat(A,0); Ax=Ser(A);

%o A[#A] = -polcoeff( subst(Ax,x, Ax - 2*Ax^2*Ax')/2, #A-1); ); H=Ax; A[n+1]}

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A219608, A259606.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Sep 06 2024