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 A291421 G.f. A(x) satisfies: A( x - A(x)^2*(1 + A(x))^2 ) = x. 1
 1, 1, 6, 45, 414, 4310, 49068, 598253, 7707738, 103981222, 1459259444, 21201220726, 317718863636, 4897066444332, 77455837982360, 1254882911977597, 20793816009974054, 351973815700006842, 6079707258590589100, 107070921557974264470, 1921112466081500096044, 35095122874748021511252, 652393778217784214993656, 12334667847853804120010726 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Compare g.f. to: C( x - C(x)^2*(1 - C(x))^2 ) = x, where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108). LINKS Paul D. Hanna, Table of n, a(n) for n = 1..300 EXAMPLE G.f.: A(x) = x + x^2 + 6*x^3 + 45*x^4 + 414*x^5 + 4310*x^6 + 49068*x^7 + 598253*x^8 + 7707738*x^9 + 103981222*x^10 + 1459259444*x^11 + 21201220726*x^12 + 317718863636*x^13 + 4897066444332*x^14 + 77455837982360*x^15 + 1254882911977597*x^16 +... such that A( x - A(x)^2 - 2*A(x)^4 - A(x)^6 ) = x. RELATED SERIES. The g.f. of A185898 equals G(x) = A(x) + A(x)^2, which begins: A(x) + A(x)^2 = x + 2*x^2 + 8*x^3 + 58*x^4 + 516*x^5 + 5264*x^6 + 59056*x^7 + 712002*x^8 + 9091360*x^9 + 121741316*x^10 +...+ A185898(n)*x^n +... and satisfies G(x - G(x)^2) = x + x^2. Also, we have the series: A(x)^2*(1 + A(x))^2 = x^2 + 4*x^3 + 20*x^4 + 148*x^5 + 1328*x^6 + 13520*x^7 + 150788*x^8 + 1804308*x^9 + 22852504*x^10 + 303523048*x^11 + 4199277144*x^12 +... where A( x - A(x)^2*(1 + A(x))^2 ) = x. Define the series reversion Ai(x) by Ai(A(x)) = x, then Ai(x) begins: Ai(x) = x - x^2 - 4*x^3 - 20*x^4 - 148*x^5 - 1328*x^6 - 13520*x^7 - 150788*x^8 - 1804308*x^9 - 22852504*x^10 +... so that Ai(x) = x - A(x)^2*(1 + A(x))^2. Finally, another series of interest is sqrt(A(x) - x) = x + 3*x^2 + 18*x^3 + 153*x^4 + 1534*x^5 + 17178*x^6 + 208276*x^7 + 2685135*x^8 + 36381426*x^9 + 513935734*x^10 + 7526074612*x^11 + 113767244374*x^12 + 1769506176124*x^13 + 28247513919396*x^14 + 461885675312008*x^15 + 7723529901763157*x^16 +... PROG (PARI) {a(n) = my(A=x, V=[1]); for(i=1, n, V=concat(V, 0); A=x*Ser(V); V[#V]=-polcoeff(subst(A, x, x - A^2*(1+A)^2), #V)); V[n]} for(n=1, 30, print1(a(n), ", ")) CROSSREFS Cf. A185898, A213591. Sequence in context: A227169 A135148 A137974 * A001879 A019577 A097814 Adjacent sequences:  A291418 A291419 A291420 * A291422 A291423 A291424 KEYWORD nonn AUTHOR Paul D. Hanna, Aug 23 2017 STATUS approved

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Last modified March 20 16:59 EDT 2018. Contains 300989 sequences. (Running on oeis4.)