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).
1, 2, 2, 3, 14, 48, 132, 406, 1430, 4952, 16796, 58416, 208012, 744468, 2674440, 9688043, 35357670, 129674822, 477638700, 1767128768, 6564120420, 24466875156, 91482563640, 343056839170, 1289904147324, 4861959143296, 18367353072152, 69533492095732, 263747951750360
Offset: 1
Keywords
Examples
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 + ... where A( A(x) - C(x) ) = x^2 and 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 + ... + A000108(n)*x^n + ... where C(x) = (1 - sqrt(1 - 4*x))/2. Let B(x) be the series reversion of A(x), B(A(x)) = x, then 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 + ... where B(x^2) = A(x) - C(x). SPECIFIC VALUES. A(-1/4) = -0.151237013399100067547709926686882225273392538412193459646... where A( A(-1/4) - (1 - sqrt(2))/2 ) = 1/16 and A(-1/4) = A(1/4) - sqrt(2)/2. A(1/4) = 0.555869767787447456853134435417966814011443399276280576942... where A( A(1/4) - 1/2 ) = 1/16. A(2/9) = 0.378446516826872823814622014107284217010617354150456751846... where A( A(2/9) - 1/3 ) = 4/81. A(3/16) = 0.28291412722740108459963161876861779881422380402719433505... where A( A(3/16) - 1/4 ) = 9/256. A(1/6) = 0.237675676844188232385878239540046791458387220170448083864... where A( A(1/6) - (1 - sqrt(1/3))/2 ) = 1/36. A(1/8) = 0.161604924202227811342812683399402861708621568115394014892... where A( A(1/8) - (1 - sqrt(1/2))/2 ) = 1/64. A(1/10) = 0.12250744402428685742299038142775672992059218375368127702... where A( A(1/10) - (1 - sqrt(3/5))/2 ) = 1/100.
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..530
Programs
-
PARI
{a(n) = my(A = x +x*O(x^n), C = serreverse(x-x^2 +x*O(x^n))); for(i=1,#binary(n), A = C + subst(serreverse(A),x,x^2)); polcoeff(A,n)} for(n=1,30, print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n)*x^n, along with the Catalan function C(x), satisfies the following formulas.
(1) A( A(x) - C(x) ) = x^2.
(2) A(x) = B(x^2) + C(x), where B(A(x)) = x (cf. A373311).
(3) A( A(x-x^2) - x ) = x^2*(1-x)^2.
(4) A(x - x^2) = x + B( x^2*(1-x)^2 ), where B(A(x)) = x.
(5) a(2*n+1) = A000108(2*n) = binomial(4*n,2*n)/(2*n+1) for n >= 0.
Comments