A371708 -id:A371708 - cp's OEIS frontend

A272483 G.f. A(x) satisfies: A( A(x)^2 ) = C(x) * A(x), where C(x) = x + C(x)^2, with A(0)=0, A'(0)=1.

1, 1, 1, 2, 7, 24, 76, 240, 787, 2670, 9233, 32293, 114051, 406588, 1461748, 5293301, 19287242, 70660178, 260127781, 961814451, 3570265304, 13299988867, 49705359457, 186309387918, 700228153534, 2638299418839, 9963349661693, 37705935306758, 142978684267052, 543164138444912, 2066978553423647, 7878398598991602, 30074161433351617, 114964340210315649

Offset: 1

Paul D. Hanna, May 03 2016

nonn

The radius of convergence of g.f. A(x) is 1/4.
Specific value S = A(1/4) = 0.4102247670209601941861161503462690608763563701332... satisfies:
(1) S = 2 * A(S^2),
(2) S = 4 * A(S^2/4) / (1 - sqrt(1 - 4*S^2)).
Limit a(n)/A000108(n-1) appears to be near 0.538...
The numerical value of this limit is 0.5377373265182445036109... . - Vaclav Kotesovec, May 07 2016

			G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 7*x^5 + 24*x^6 + 76*x^7 + 240*x^8 + 787*x^9 + 2670*x^10 + 9233*x^11 + 32293*x^12 +...
such that A( A(x)^2 ) = C(x) * A(x), where C(x) = x + C(x)^2.
RELATED SERIES.
C(x) = x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + 429*x^8 + 1430*x^9 + 4862*x^10 + 16796*x^11 + 58786*x^12 +...+ A000108(n-1)*x^n +...
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 19*x^6 + 66*x^7 + 218*x^8 + 708*x^9 + 2351*x^10 + 8034*x^11 + 27980*x^12 + 98548*x^13 + 350148*x^14 +...
A( A(x)^2 ) = x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 30*x^6 + 96*x^7 + 312*x^8 + 1030*x^9 + 3472*x^10 + 11932*x^11 + 41619*x^12 + 146828*x^13 + 522914*x^14 +...
where A( A(x)^2 ) = C(x)*A(x).
A(x-x^2) = x - x^3 + 2*x^5 - 6*x^7 + 18*x^9 - 59*x^11 + 204*x^13 - 728*x^15 + 2672*x^17 - 10022*x^19 + 38243*x^21 - 148039*x^23 + 579954*x^25 +...
where A(x-x^2) = Series_Reversion( A(x^2)/x ).
A( A(x-x^2)^2 ) = x^2 - x^4 + 2*x^6 - 6*x^8 + 18*x^10 - 59*x^12 + 204*x^14 - 728*x^16 + 2672*x^18 - 10022*x^20 + 38243*x^22 - 148039*x^24 +...
where A( A(x-x^2)^2 ) = x*A(x-x^2).

Paul D. Hanna, Table of n, a(n) for n = 1..300

Cf. A272484, A371708.

{a(n) = my(A=x,C=x,X=x+x*O(x^n)); for(i=1,n, C = X + C^2; A = (2*A - subst(A,x,A^2)/C )); polcoeff(A,n)}
for(n=1,40,print1(a(n),", "))

G.f. A(x) satisfies:
(1) A( A(x-x^2)^2 ) = x * A(x-x^2).
(2) A(x-x^2) = Series_Reversion( A(x^2)/x ).
(3) A( A(x^2)/x - A(x^2)^2/x^2 ) = x.
a(n) ~ c * 4^n / n^(3/2), where c = 0.075846449576603052427... . - Vaclav Kotesovec, May 07 2016

A372528 Expansion of g.f. A(x) satisfying A( -x * A( x - x^2 ) ) = -x^2.

Original entry on oeis.org

1, 1, 3, 8, 22, 65, 200, 637, 2090, 7021, 24041, 83611, 294511, 1048376, 3765080, 13623820, 49617990, 181733222, 668947823, 2473277248, 9180700787, 34200489886, 127819746470, 479124333321, 1800838945043, 6785517883825, 25626477179000, 96988079848223, 367794448974300, 1397301289617580

Offset: 1

Paul D. Hanna, May 05 2024

nonn

			G.f.: A(x) = x + x^2 + 3*x^3 + 8*x^4 + 22*x^5 + 65*x^6 + 200*x^7 + 637*x^8 + 2090*x^9 + 7021*x^10 + 24041*x^11 + 83611*x^12 + ...
RELATED SERIES.
Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then
R(x) = x - x^2 - x^3 + 2*x^4 + 4*x^5 - 9*x^6 - 18*x^7 + 44*x^8 + 91*x^9 - 234*x^10 - 496*x^11 + 1318*x^12 + ...
where A(x - x^2) = R(-x^2)/(-x).
Also, the bisections B1 and B2 of R(x) are
B1 = (R(x) - R(-x))/2 = x - x^3 + 4*x^5 - 18*x^7 + 91*x^9 - 496*x^11 + 2839*x^13 - 16836*x^15 + 102545*x^17 - 637733*x^19 + ...
B2 = (R(x) + R(-x))/2 = -x^2 + 2*x^4 - 9*x^6 + 44*x^8 - 234*x^10 + 1318*x^12 - 7722*x^14 + 46594*x^16 - 287611*x^18 + 1807720*x^20 + ...
and satisfy B1^2 + B2 = 0 and A(-x*B1) = -B1^2.
SPECIFIC VALUES.
A( -A(2/9) / 3 ) = -1/9 where
A(2/9) = 0.3655811677545134614272600644874552972994602150418984...
A( -A(3/16) / 4 ) = -1/16 where
A(3/16) = 0.2645434685642398513217156896362957133168212272114320...
A( -A(4/25) / 5 ) = -1/25 where
A(4/25) = 0.2076566162630115730635446744577181791494166261819659...
A( -A(5/36) / 6 ) = -1/36 where
A(5/36) = 0.1711609712404346976409014231532840797963445277760447...

Paul D. Hanna, Table of n, a(n) for n = 1..520

Cf. A371708, A000108.

{a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( x^2 + subst(Ser(A), x, -x*subst(Ser(A), x, x - x^2) ), #A)); A[n+1]}
for(n=1, 35, print1(a(n), ", "))

G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with its series reversion R(x), satisfy the following formulas.
(1) A( -x*A(x - x^2) ) = -x^2.
(2) A(x - x^2) = R(-x^2)/(-x).
(3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.
(4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).
(5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).
(6) A(x) = -A(x - 2*C(x)) where C(x) = -C(x - 2*C(x)) is a g.f. of the Catalan numbers (A000108).
(7) A( -A(x)*C(x) ) = -C(x)^2 where C(x) = (1 - sqrt(1 - 4*x))/2 is a g.f. of the Catalan numbers (A000108).

cp's OEIS Frontend

A272483 G.f. A(x) satisfies: A( A(x)^2 ) = C(x) * A(x), where C(x) = x + C(x)^2, with A(0)=0, A'(0)=1.

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