A326553 - OEIS

%I #3 Jul 26 2019 22:08:49

%S 1,3,19,189,2521,41813,826099,18889209,489450841,14142037665,

%T 449838952795,15587619260117,583210749984409,23380025164256829,

%U 997331252199242083,44981225190596283921,2131771651553974931929,105498794931132306727769,5415121471957856287745611,286003536194104676920620909,15383978962252356122447259865

%N E.g.f. A(x) satisfies: Sum_{n>=1} (-1)^(n-1) * A(x)^n / (n + A(x)^n) = x.

%F E.g.f.: Series_Reversion( Sum_{n>=1} (-1)^(n-1) * x^n / (n + x^n) ).

%e E.g.f.: A(x) = x + 3*x^2/2! + 19*x^3/3! + 189*x^4/4! + 2521*x^5/5! + 41813*x^6/6! + 826099*x^7/7! + 18889209*x^8/8! + 489450841*x^9/9! + 14142037665*x^10/10! + 449838952795*x^11/11! + ...

%e such that

%e x = A(x)/(1 + A(x)) + A(x)^2/(2 + A(x)^2) + A(x)^3/(3 + A(x)^3) + A(x)^4/(4 + A(x)^4) + A(x)^5/(5 + A(x)^5) + A(x)^6/(6 + A(x)^6) + ...

%e The series reversion of the e.g.f. begins:

%e Series_Reversion(A(x)) = x - 3*x^2/2! + 8*x^3/3! - 24*x^4/4! + 144*x^5/5! - 1010*x^6/6! + 5760*x^7/7! - 40320*x^8/8! + 416640*x^9/9! - 4250232*x^10/10! + 43545600*x^11/11! - 511526400*x^12/12! + ...

%e which equals Sum_{n>=1} (-1)^(n-1) * x^n / (n + x^n).

%o (PARI) {a(n) = my(A = serreverse( sum(m=1,n+1, (-1)^(m-1)*x^m/(m + x^m +x*O(x^n))) ));

%o n!*polcoeff(H=A,n)}

%o for(n=1,25,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jul 26 2019