A326553 E.g.f. A(x) satisfies: Sum_{n>=1} (-1)^(n-1) * A(x)^n / (n + A(x)^n) = x.

1, 3, 19, 189, 2521, 41813, 826099, 18889209, 489450841, 14142037665, 449838952795, 15587619260117, 583210749984409, 23380025164256829, 997331252199242083, 44981225190596283921, 2131771651553974931929, 105498794931132306727769, 5415121471957856287745611, 286003536194104676920620909, 15383978962252356122447259865

Offset

1,2

Links

Table of n, a(n) for n=1..21.

Formula

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

Example

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! + ...
such that
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) + ...
The series reversion of the e.g.f. begins:
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! + ...
which equals Sum_{n>=1} (-1)^(n-1) * x^n / (n + x^n).

Prog

(PARI) {a(n) = my(A = serreverse( sum(m=1, n+1, (-1)^(m-1)*x^m/(m + x^m +x*O(x^n))) ));
n!*polcoeff(H=A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A202617 A143633 A367180 * A052888 A141623 A229234

Adjacent sequences: A326550 A326551 A326552 * A326554 A326555 A326556

Keyword

nonn

Author

Paul D. Hanna, Jul 26 2019

Status

approved