A299425 E.g.f. A(x) satisfies: 1 - x = Sum_{n>=0} (Integral -A(x)^n dx)^n / n!.

1, 1, 5, 43, 514, 7778, 141427, 2990741, 71982197, 1943318293, 58274530808, 1928170445532, 70050754418969, 2781721368620685, 120104488614903489, 5601368827311472211, 280010977440133986954, 14884513825530929143826, 835402492859596917623243, 49259935601350475622014329, 3045801421611820426020858069

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..100

Formula

E.g.f. A(x) satisfies:

(1) 1 - x = Sum_{n>=0} (Integral -A(x)^n dx)^n / n!.

(2) 1 = Sum_{n>=0} A(x)^(n+1) * (Integral -A(x)^(n+1) dx)^n / n!.

Example

E.g.f.: A(x) = 1 + x + 5*x^2/2! + 43*x^3/3! + 514*x^4/4! + 7778*x^5/5! + 141427*x^6/6! + 2990741*x^7/7! + 71982197*x^8/8! + 1943318293*x^9/9! + 58274530808*x^10/10! + ...
such that
1 - x = 1 - (Integral A(x) dx) + (Integral A(x)^2 dx)^2/2! - (Integral A(x)^3 dx)^3/3! + (Integral A(x)^4 dx)^4/4! - (Integral A(x)^5 dx)^5/5! + ...
also
1 = A(x) - A(x)^2*(Integral A(x)^2 dx) + A(x)^3*(Integral A(x)^3 dx)^2/2! - A(x)^4*(Integral A(x)^4 dx)^3/3! + A(x)^5*(Integral A(x)^5 dx)^4/4! + ...
Related series.
A'(x) = F(x)/G(x) where
F(x) = Sum_{n>=0} A(x)^(2*n+4) * (Integral -A(x)^(n+2) dx)^n / n! and
G(x) = Sum_{n>=0} (n+1) * A(x)^n * (Integral -A(x)^(n+1) dx)^n / n!.

Prog

(PARI) {a(n) = my(A=1); for(i=1, n, A = 1 - sum(m=1, n, A^(m+1) * intformal( -A^(m+1) +x*O(x^n) )^m/m!)); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = -Vec( sum(m=0, #A+1, intformal( -Ser(A)^(m+1) )^m/m! * Ser(A)^(m+1)) )[#A] ); n!*A[n+1]}
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Sequence in context: A188365 A107720 A362188 * A360618 A060053 A227176

Adjacent sequences: A299422 A299423 A299424 * A299426 A299427 A299428

Keyword

sign

Author

Paul D. Hanna, Feb 15 2018

Status

approved