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 A214372 G.f. satisfies: A(x) = x + A(x)^2*(1 + A(x))^2. 3
 1, 1, 4, 16, 74, 364, 1876, 9993, 54582, 304040, 1720576, 9864114, 57169168, 334404892, 1971641096, 11705119157, 69911438676, 419798442416, 2532791105844, 15346498242460, 93344296450020, 569741502841020, 3488539758461880, 21422341792366320 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 LINKS G. C. Greubel, Table of n, a(n) for n = 1..1000 ElÅ¼bieta Liszewska, Wojciech MÅ‚otkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019. FORMULA G.f. A(x) satisfies: (1) A(x) = Series_Reversion( x - x^2*(1+x)^2 ). (2) A(x) = x + Series_Reversion( (sqrt(1+4*x) - 1)/2 - x^2 )^2. - Paul D. Hanna, Oct 24 2013 (3) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n)*(1+x)^(2*n)/n!. (4) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) x^(2*n-1)*(1+x)^(2*n)/n! ). Recurrence: 31*(n-2)*(n-1)*n*(104*n - 293)*a(n) = 8*(n-2)*(n-1)*(1768*n^2 - 7633*n + 7425)*a(n-1) + 20*(n-2)*(2080*n^3 - 14180*n^2 + 31612*n - 22995)*a(n-2) + 8*(2*n-5)*(4*n-13)*(4*n-11)*(104*n - 189)*a(n-3). - Vaclav Kotesovec, Sep 17 2013 a(n) ~ 1/312*sqrt(78)*sqrt((26533 + 50583*sqrt(78))^(2/3) - 5837 + 13*(26533 + 50583*sqrt(78))^(1/3))/((26533 + 50583*sqrt(78))^(1/6)) * (4/93*((209773 + 4836*sqrt(78))^(2/3) + 3481 + 34*(209773 + 4836* sqrt(78))^(1/3))/(209773 + 4836*sqrt(78))^(1/3))^n / (n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Sep 17 2013 EXAMPLE G.f.: A(x) = x + x^2 + 4*x^3 + 16*x^4 + 74*x^5 + 364*x^6 + 1876*x^7 +... Related expansions: A(x) = x + x^2*(1+x)^2 + d/dx x^4*(1+x)^4/2! + d^2/dx^2 x^6*(1+x)^6/3! + d^3/dx^3 x^8*(1+x)^8/4! +... log(A(x)/x) = x*(1+x)^2 + d/dx x^3*(1+x)^4/2! + d^2/dx^2 x^5*(1+x)^6/3! + d^3/dx^3 x^7*(1+x)^8/4! +... A(x)^2 = x^2 + 2*x^3 + 9*x^4 + 40*x^5 + 196*x^6 + 1004*x^7 + 5328*x^8 +... (1+A(x))^2 = 1 + 2*x + 3*x^2 + 10*x^3 + 41*x^4 + 188*x^5 + 924*x^6 + 4756*x^7 + 25314*x^8 +... Series reversion shows a relation to the Catalan numbers (A000108): Series_Reversion( sqrt(A(x) - x) ) =  (sqrt(1+4*x) - 1)/2 - x^2, which begins x - 2*x^2 + 2*x^3 - 5*x^4 + 14*x^5 - 42*x^6 + 132*x^7 - 429*x^8 +... where sqrt(A(x) - x) = x + 2*x^2 + 6*x^3 + 25*x^4 + 114*x^5 + 560*x^6 + 2880*x^7 +...+ A229042(n)*x^n +... MAPLE a:= n-> coeff(series(RootOf(A=x+A^2*(1+A)^2, A), x, n+1), x, n): seq(a(n), n=1..30);  # Alois P. Heinz, Feb 13 2017 MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x - x^2*(1+x)^2, {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Sep 17 2013 *) PROG (PARI) {a(n)=polcoeff(serreverse(x-x^2*(1+x)^2+x*O(x^n)), n)} for(n=1, 30, print1(a(n), ", ")) (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, x^(2*m)*(1+x+x*O(x^n))^(2*m)/m!)); polcoeff(A, n)} (PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D} {a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, x^(2*m-1)*(1+x+x*O(x^n))^(2*m)/m!))); polcoeff(A, n)} (PARI) {a(n)=polcoeff(x + serreverse( (sqrt(1+4*x +x*O(x^n)) - 1)/2 - x^2 )^2, n)} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A229042. Sequence in context: A091530 A344248 A301455 * A109957 A306495 A331159 Adjacent sequences:  A214369 A214370 A214371 * A214373 A214374 A214375 KEYWORD nonn AUTHOR Paul D. Hanna, Jul 14 2012 STATUS approved

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Last modified August 3 01:55 EDT 2021. Contains 346429 sequences. (Running on oeis4.)