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A341377 E.g.f. A(x) satisfies: A(x) = P(x)/Q(x) where P(x) = Sum_{n>=0} (2^n*x^n/n!) * exp(x*A(x)^n) and Q(x) = Sum_{n>=0} (x^n/n!) * exp(x*A(x)^n). 0
1, 1, 1, 7, 37, 501, 5491, 100423, 1750729, 40959145, 983720071, 28320815931, 862784739037, 29728071447517, 1093233588351451, 44072354567202991, 1894875749109296401, 87722973117476411985, 4313507974583583130255, 225835795120784011395427 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Table of n, a(n) for n=0..19.

FORMULA

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

(1) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (2^n*x^n/n!) * exp(x*A(x)^n) and

Q(x) = Sum_{n>=0} (x^n/n!) * exp(x*A(x)^n).

(2) A(x) = P(x)/Q(x) where

P(x) = Sum_{n>=0} (x^n/n!) * exp(2*x*A(x)^n) and

Q(x) = Sum_{n>=0} (x^n/n!) * exp(x*A(x)^n).

EXAMPLE

E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 37*x^4/4! + 501*x^5/5! + 5491*x^6/6! + 100423*x^7/7! + 1750729*x^8/8! + 40959145*x^9/9! + 983720071*x^10/10! + ...

such that A(x) = P(x)/Q(x) where

P(x) = exp(x) + 2*x*exp(x*A(x)) + 2^2*x^2*exp(x*A(x)^2)/2! + 2^3*x^3*exp(x*A(x)^3)/3! + 2^4*x^4*exp(x*A(x)^4)/4! + 2^5*x^5*exp(x*A(x)^5)/5! + ...

Q(x) = exp(x) + x*exp(x*A(x)) + x^2*exp(x*A(x)^2)/2! + x^3*exp(x*A(x)^3)/3! + x^4*exp(x*A(x)^4)/4! + x^5*exp(x*A(x)^5)/5! + ...

also

P(x) = exp(x) + x*exp(2*x*A(x)) + x^2*exp(2*x*A(x)^2)/2! + x^3*exp(2*x*A(x)^3)/3! + x^4*exp(2*x*A(x)^4)/4! + x^5*exp(2*x*A(x)^5)/5! + ...

explicitly,

P(x) = 1 + 3*x + 9*x^2/2! + 39*x^3/3! + 249*x^4/4! + 2323*x^5/5! + 27789*x^6/6! + 421851*x^7/7! + 7577425*x^8/8! + 160651683*x^9/9! + 3868078869*x^10/10! + ...

Q(x) = 1 + 2*x + 4*x^2/2! + 14*x^3/3! + 76*x^4/4! + 652*x^5/5! + 7054*x^6/6! + 102650*x^7/7! + 1741016*x^8/8! + 35911592*x^9/9! + 832884154*x^10/10! + ...

PROG

(PARI) {a(n) = my(A=1 +x*O(x^n), P=1, Q=1); for(i=1, n,

P = sum(m=0, n, 2^m*x^m/m! * exp(x*A^m +x*O(x^n)) );

Q = sum(m=0, n, x^m/m! * exp(x*A^m +x*O(x^n)) );

A = P/Q); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n) = my(A=1 +x*O(x^n), P=1, Q=1); for(i=1, n,

P = sum(m=0, n, x^m/m! * exp(2*x*A^m +x*O(x^n)) );

Q = sum(m=0, n, x^m/m! * exp(x*A^m +x*O(x^n)) );

A = P/Q); n!*polcoeff(A, n)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Sequence in context: A330388 A276231 A097493 * A082687 A117731 A155010

Adjacent sequences:  A341374 A341375 A341376 * A341378 A341379 A341380

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Feb 22 2021

STATUS

approved

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Last modified April 16 23:40 EDT 2021. Contains 343051 sequences. (Running on oeis4.)