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 A120010 G.f.: A(x) = (1-sqrt(1-4*x))/2 o x/(1-x) o (x-x^2), a composition of functions involving the Catalan function and its inverse. 9
 1, 1, 1, 2, 6, 18, 53, 158, 481, 1491, 4688, 14913, 47913, 155261, 506881, 1665643, 5504988, 18287338, 61027991, 204499397, 687808931, 2321177071, 7857504876, 26673769002, 90783820081, 309720079813, 1058984020333, 3628267267358 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS The n-th iteration of g.f. A(x) is: (1-sqrt(1-4*x))/2 o x/(1-n*x) o (x-x^2) = (1 - sqrt(1 - 4*(x-x^2)/(1-n*x+n*x^2) ))/2. See A120009 for the transpose of the composition of the same functions. Row sums of A155839. [From Paul Barry, Jan 28 2009] LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..200 FORMULA G.f.: A(x) = (1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2) ))/2. a(n)=sum{k=0..n, sum{j=0..n, (-1)^(n-j)*C(j+1,n-j)*C(j,k)*if(k<=j, A000108(j-k),0)}}. [offset 0]. [From Paul Barry, Jan 28 2009] Conjecture: n*a(n) +2*(4-3*n)*a(n-1) +(11*n-26)*a(n-2) +10*(3-n)*a(n-3) +5*(n-4)*a(n-4)= 0. - R. J. Mathar, Nov 14 2011 a(n) ~ sqrt(5*sqrt(5)-5) * (5+sqrt(5))^n / (sqrt(Pi) * n^(3/2) * 2^(n+7/2)). - Vaclav Kotesovec, Feb 13 2014 EXAMPLE G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 18*x^6 + 53*x^7 + 158*x^8 +... MATHEMATICA Rest[CoefficientList[Series[(1-Sqrt[1-4*(x-x^2)/(1-x+x^2)])/2, {x, 0, 20}], x]] (* Vaclav Kotesovec, Feb 13 2014 *) PROG (PARI) {a(n)=polcoeff((1 - sqrt(1 - 4*(x-x^2)/(1-x+x^2+x*O(x^n)) ))/2, n)} for(n=1, 35, print1(a(n), ", ")) CROSSREFS Cf. A120009 (composition transpose), A000108 (Catalan). Sequence in context: A005507 A252822 A094864 * A132790 A214799 A072850 Adjacent sequences:  A120007 A120008 A120009 * A120011 A120012 A120013 KEYWORD nonn AUTHOR Paul D. Hanna, Jun 03 2006 STATUS approved

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Last modified August 15 13:11 EDT 2020. Contains 336504 sequences. (Running on oeis4.)