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A373311 Expansion of g.f. A(x) satisfying A( A(x^2) + C(x) ) = x, where C(x) = x + C(x)^2 is the Catalan function (A000108).

%I A373311 #6 Jul 07 2024 20:23:37
%S A373311 1,-2,6,-23,90,-370,1568,-6802,30032,-134422,608136,-2774480,12741844,
%T A373311 -58820272,272617696,-1267318854,5904152560,-27545442786,128610298720,
%U A373311 -600579852616,2803385644716,-13072680693872,60864695165024,-282763138097520,1309945271204312,-6047009494466692
%N A373311 Expansion of g.f. A(x) satisfying A( A(x^2) + C(x) ) = x, where C(x) = x + C(x)^2 is the Catalan function (A000108).
%H A373311 Paul D. Hanna, <a href="/A373311/b373311.txt">Table of n, a(n) for n = 1..530</a>
%F A373311 G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with the Catalan function C(x), satisfies the following formulas.
%F A373311 (1) A( A(x^2) + C(x) ) = x.
%F A373311 (2) A(x^2) = B(x) - C(x), where B(A(x)) = x (cf. A373310).
%F A373311 (3) A( x + A(x^2*(1-x)^2) ) = x - x^2.
%F A373311 (4) A( x^2*(1 - x)^2 ) = B(x-x^2) - x, where B(A(x)) = x.
%e A373311 G.f.: A(x) = x - 2*x^2 + 6*x^3 - 23*x^4 + 90*x^5 - 370*x^6 + 1568*x^7 - 6802*x^8 + 30032*x^9 - 134422*x^10 + 608136*x^11 - 2774480*x^12 + ...
%e A373311 where A( A(x^2) + C(x) ) = x and C(x) = x + C(x)^2.
%e A373311 RELATED SERIES.
%e A373311 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 + ... + A000108(n)*x^n + ...
%e A373311 where C(x) = (1 - sqrt(1 - 4*x))/2.
%e A373311 Let B(x) be the series reversion of A(x), B(A(x)) = x, then
%e A373311 B(x) = x + 2*x^2 + 2*x^3 + 3*x^4 + 14*x^5 + 48*x^6 + 132*x^7 + 406*x^8 + 1430*x^9 + 4952*x^10 + ... + A373310(n)*x^n + ...
%e A373311 where B(x) = A(x^2) + C(x).
%e A373311 SPECIFIC VALUES.
%e A373311 A(1/8^2) = 0.0151583147955015735432348644518273813510395283898972054...
%e A373311 where A( A(1/8^2) + (1 - sqrt(1/2))/2 ) = 1/8.
%e A373311 A(1/10^2) = 0.009805778645028545940916921405996691003884354282840359...
%e A373311 where A( A(1/10^2) + (1 - sqrt(3/5))/2 ) = 1/10.
%e A373311 A(1/6) = 0.12804766097193434952321787018897962946429947564936619827...
%e A373311 A(1/7) = 0.11334295578613203758002178169147530906130640897889608109...
%e A373311 A(1/8) = 0.10167460278631364378329951659088472698712642799619893229...
%e A373311 A(1/16) = 0.0558697677874474568531344354179668140114433992762805769...
%o A373311 (PARI) {a(n) = my(A = x +x*O(x^n), C = serreverse(x-x^2 +x*O(x^n)));
%o A373311 for(i=1,#binary(n), A = serreverse(subst(A,x,x^2) + C)); polcoeff(A,n)}
%o A373311 for(n=1,30, print1(a(n),", "))
%Y A373311 Cf. A373310, A000108.
%K A373311 sign
%O A373311 1,2
%A A373311 _Paul D. Hanna_, Jul 07 2024