login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A374572
Expansion of g.f. A(x) satisfying A(x)^2 = A( x*(1+x)*A(x) ).
1
1, 1, 1, 3, 5, 11, 27, 69, 183, 481, 1283, 3453, 9361, 25651, 70927, 197721, 555039, 1567345, 4449023, 12686465, 36323203, 104381397, 300958959, 870378337, 2524129349, 7338679127, 21386456807, 62459196233, 182776933033, 535861013939, 1573742036447, 4629306941913
OFFSET
1,4
LINKS
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x)^2 = A( x*(1+x)*A(x) ).
(2) A(x)^4 = A( x*(1+x)*A(x)^3 + x^2*(1+x)^2*A(x)^4 ).
(3) A(x)^8 = A( x*(1+x)*A(x)^7 + x^2*(1+x)^2*A(x)^8 + x^2*(1+x)^2*A(x)^10 + 2*x^3*(1+x)^3*A(x)^11 + x^4*(1+x)^4*A(x)^12 ).
(4) x^2 = A( x*B(x)*(1 + B(x)) ) where A(B(x)) = x.
a(n) ~ c * d^n / n^(3/2), where d = 3.0870367560295429... and c = 0.17761867899908... - Vaclav Kotesovec, Jul 19 2024
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 5*x^5 + 11*x^6 + 27*x^7 + 69*x^8 + 183*x^9 + 481*x^10 + 1283*x^11 + 3453*x^12 + 9361*x^13 + 25651*x^14 + 70927*x^15 + ...
where A(x)^2 = A( x*(1+x)*A(x) ).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 8*x^5 + 17*x^6 + 38*x^7 + 95*x^8 + 244*x^9 + 649*x^10 + 1738*x^11 + 4699*x^12 + ...
Let B(x) be the series reversion of g.f. A(x), B(A(x)) = x, then
B(x) = x - x^2 + x^3 - 3*x^4 + 9*x^5 - 25*x^6 + 71*x^7 - 219*x^8 + 693*x^9 - 2197*x^10 + 7069*x^11 - 23135*x^12 + ...
where B(x^2) = x*B(x)*(1 + B(x)).
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.301949314609828865985839329094529550482897401344979...
where 1/4 = A( t*(1 + t)/2 ).
A(3/10) = 0.492388112365452715229250795508017422919418907801551...
where A(3/10)^2 = A( (39/100)*A(3/10) ).
A(2/7) = 0.443877424659041232765055763766392304444609934055603...
where A(2/7)^2 = A( (18/49)*A(2/7) ).
A(1/4) = 0.352241294433584221893793757577235288109595399125986...
where A(1/4)^2 = A( (5/16)*A(1/4) ).
A(1/5) = 0.255826785620580342641277164817159026900345909888978...
where A(1/5)^2 = A( (6/25)*A(1/5) ).
PROG
(PARI) {a(n) = my(A=[0, 1], Ax); for(i=1, n, A = concat(A, 0); Ax = Ser(A);
A[#A] = polcoeff( subst(Ax, x, x*(1+x)*Ax ) - Ax^2, #A) ); A[n+1]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
Cf. A075864.
Sequence in context: A369344 A204857 A292855 * A265941 A372099 A333629
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 18 2024
STATUS
approved