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URL: https://oeis.org/A341377

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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
OFFSET
0,4
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: A371710 A276231 A097493 * A082687 A117731 A389783
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 22 2021
STATUS
approved