cp's OEIS Frontend

A373310 Expansion of g.f. A(x) satisfying A( A(x) - C(x) ) = x^2, where C(x) = x + C(x)^2 is the Catalan function (A000108).

%I A373310 #10 Jul 07 2024 20:23:29
%S A373310 1,2,2,3,14,48,132,406,1430,4952,16796,58416,208012,744468,2674440,
%T A373310 9688043,35357670,129674822,477638700,1767128768,6564120420,
%U A373310 24466875156,91482563640,343056839170,1289904147324,4861959143296,18367353072152,69533492095732,263747951750360
%N A373310 Expansion of g.f. A(x) satisfying A( A(x) - C(x) ) = x^2, where C(x) = x + C(x)^2 is the Catalan function (A000108).
%C A373310 When is a(n) odd? It appears that a(n) is odd iff n = 2^A279430(m) for some m >= 1.
%H A373310 Paul D. Hanna, <a href="/A373310/b373310.txt">Table of n, a(n) for n = 1..530</a>
%F A373310 G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with the Catalan function C(x), satisfies the following formulas.
%F A373310 (1) A( A(x) - C(x) ) = x^2.
%F A373310 (2) A(x) = B(x^2) + C(x), where B(A(x)) = x (cf. A373311).
%F A373310 (3) A( A(x-x^2) - x ) = x^2*(1-x)^2.
%F A373310 (4) A(x - x^2) = x + B( x^2*(1-x)^2 ), where B(A(x)) = x.
%F A373310 (5) a(2*n+1) = A000108(2*n) = binomial(4*n,2*n)/(2*n+1) for n >= 0.
%e A373310 G.f.: A(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 + 16796*x^11 + 58416*x^12 + 208012*x^13 + 744468*x^14 + ...
%e A373310 where A( A(x) - C(x) ) = x^2 and C(x) = x + C(x)^2.
%e A373310 RELATED SERIES.
%e A373310 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 A373310 where C(x) = (1 - sqrt(1 - 4*x))/2.
%e A373310 Let B(x) be the series reversion of A(x), B(A(x)) = x, then
%e A373310 B(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 + ... + A373311(n)*x^n + ...
%e A373310 where B(x^2) = A(x) - C(x).
%e A373310 SPECIFIC VALUES.
%e A373310 A(-1/4) = -0.151237013399100067547709926686882225273392538412193459646...
%e A373310 where A( A(-1/4) - (1 - sqrt(2))/2 ) = 1/16
%e A373310 and A(-1/4) = A(1/4) - sqrt(2)/2.
%e A373310 A(1/4) = 0.555869767787447456853134435417966814011443399276280576942...
%e A373310 where A( A(1/4) - 1/2 ) = 1/16.
%e A373310 A(2/9) = 0.378446516826872823814622014107284217010617354150456751846...
%e A373310 where A( A(2/9) - 1/3 ) = 4/81.
%e A373310 A(3/16) = 0.28291412722740108459963161876861779881422380402719433505...
%e A373310 where A( A(3/16) - 1/4 ) = 9/256.
%e A373310 A(1/6) = 0.237675676844188232385878239540046791458387220170448083864...
%e A373310 where A( A(1/6) - (1 - sqrt(1/3))/2 ) = 1/36.
%e A373310 A(1/8) = 0.161604924202227811342812683399402861708621568115394014892...
%e A373310 where A( A(1/8) - (1 - sqrt(1/2))/2 ) = 1/64.
%e A373310 A(1/10) = 0.12250744402428685742299038142775672992059218375368127702...
%e A373310 where A( A(1/10) - (1 - sqrt(3/5))/2 ) = 1/100.
%o A373310 (PARI) {a(n) = my(A = x +x*O(x^n), C = serreverse(x-x^2 +x*O(x^n)));
%o A373310 for(i=1,#binary(n), A = C + subst(serreverse(A),x,x^2)); polcoeff(A,n)}
%o A373310 for(n=1,30, print1(a(n),", "))
%Y A373310 Cf. A373311, A000108.
%K A373310 nonn
%O A373310 1,2
%A A373310 _Paul D. Hanna_, Jul 07 2024