%I #15 May 05 2024 19:57:22

%S 1,1,3,8,22,65,200,637,2090,7021,24041,83611,294511,1048376,3765080,

%T 13623820,49617990,181733222,668947823,2473277248,9180700787,

%U 34200489886,127819746470,479124333321,1800838945043,6785517883825,25626477179000,96988079848223,367794448974300,1397301289617580

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

%H Paul D. Hanna, <a href="/A372528/b372528.txt">Table of n, a(n) for n = 1..520</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with its series reversion R(x), satisfy the following formulas.

%F (1) A( -x*A(x - x^2) ) = -x^2.

%F (2) A(x - x^2) = R(-x^2)/(-x).

%F (3) (R(x) - R(-x))^2 + 2*(R(x) + R(-x)) = 0.

%F (4) R(x) = R(-x) - 1 + sqrt(1 - 4*R(-x)).

%F (5) A(x) = -A( x - 1 + sqrt(1 - 4*x) ).

%F (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).

%F (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).

%e 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 + ...

%e RELATED SERIES.

%e Let R(x) be the series reversion of g.f. A(x), R(A(x)) = x, then

%e 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 + ...

%e where A(x - x^2) = R(-x^2)/(-x).

%e Also, the bisections B1 and B2 of R(x) are

%e 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 + ...

%e 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 + ...

%e and satisfy B1^2 + B2 = 0 and A(-x*B1) = -B1^2.

%e SPECIFIC VALUES.

%e A( -A(2/9) / 3 ) = -1/9 where

%e A(2/9) = 0.3655811677545134614272600644874552972994602150418984...

%e A( -A(3/16) / 4 ) = -1/16 where

%e A(3/16) = 0.2645434685642398513217156896362957133168212272114320...

%e A( -A(4/25) / 5 ) = -1/25 where

%e A(4/25) = 0.2076566162630115730635446744577181791494166261819659...

%e A( -A(5/36) / 6 ) = -1/36 where

%e A(5/36) = 0.1711609712404346976409014231532840797963445277760447...

%o (PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);

%o A[#A] = polcoeff( x^2 + subst(Ser(A), x, -x*subst(Ser(A), x, x - x^2) ), #A)); A[n+1]}

%o for(n=1, 35, print1(a(n), ", "))

%Y Cf. A371708, A000108.

%K nonn,new

%O 1,3

%A _Paul D. Hanna_, May 05 2024