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A275756 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^6 ).

%I A275756 #16 Aug 20 2016 22:56:52
%S A275756 1,1,2,6,20,71,264,1018,4032,16305,67042,279444,1178088,5014596,
%T A275756 21521488,93027025,404630318,1769704106,7778030834,34335337802,
%U A275756 152168657438,676796514510,3019945599904,13515300673984,60649985907334,272847379282493,1230295797205452,5559373120441048,25171114275512520,114177375142080814,518806321789317040,2361183952087172306,10762422470020855820,49125407360603361370,224533932290057629076,1027553322543206612019,4708070541211739962738,21595828228486254332762
%N A275756 G.f. satisfies: A(x) = x + A( A(x)^2 - A(x)^6 ).
%C A275756 Compare to: C(x) = x + C( C(x)^2 - C(x)^4 ) holds when C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).
%C A275756 Compare to: C(x) = x + C( C(x)^3 - C(x)^9 ) holds when C(x) = x + C(x)^3 is a g.f. of the ternary tree numbers (A001764).
%H A275756 Paul D. Hanna, <a href="/A275756/b275756.txt">Table of n, a(n) for n = 1..300</a>
%F A275756 G.f. satisfies:
%F A275756 (1) A(x - A(x^2 - x^6)) = x.
%F A275756 (2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1) / (n+1)!.
%F A275756 (3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^2-x^6)^(n+1)/x / (n+1)! ).
%e A275756 G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 264*x^7 + 1018*x^8 + 4032*x^9 + 16305*x^10 + 67042*x^11 + 279444*x^12 + 1178088*x^13 + 5014596*x^14 +...
%e A275756 such that A(x) = x + A( A(x)^2 - A(x)^6 ).
%e A275756 RELATED SERIES.
%e A275756 A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 16*x^5 + 56*x^6 + 206*x^7 + 786*x^8 + 3088*x^9 + 12408*x^10 + 50754*x^11 + 210639*x^12 + 884784*x^13 + 3754424*x^14 +...
%e A275756 A(x)^6 = x^6 + 6*x^7 + 27*x^8 + 116*x^9 + 495*x^10 + 2112*x^11 + 9035*x^12 + 38820*x^13 + 167628*x^14 + 727480*x^15 + 3172455*x^16 +...
%e A275756 A(x^2 - x^6) = x^2 + x^4 + x^6 + 4*x^8 + 14*x^10 + 48*x^12 + 170*x^14 + 628*x^16 + 2382*x^18 + 9202*x^20 + 36098*x^22 + 143484*x^24 + 576638*x^26 + 2339050*x^28 +...
%e A275756 where Series_Reversion(A(x)) = x - A(x^2 - x^6).
%o A275756 (PARI) {a(n) = my(A=x); for(i=1,n, A = x + subst(A,x, A^2 - A^6 +x*O(x^n))); polcoeff(A,n)}
%o A275756 for(n=1,40,print1(a(n),", "))
%Y A275756 Cf. A275755, A275757, A275758, A141200.
%K A275756 nonn
%O A275756 1,3
%A A275756 _Paul D. Hanna_, Aug 20 2016