%I #15 Aug 22 2016 23:15:01
%S 1,1,4,21,126,817,5574,39418,286286,2122491,15995696,122166551,
%T 943430560,7353998931,57783603764,457176705018,3639000808140,
%U 29119701312548,234120338807316,1890257713736568,15319612051101438,124583720191974904,1016307862050772614,8314217332992596050,68193993494598345010,560671685990956975367,4619857060146629819160,38144728242794104501561,315546193363448088862064,2614910268303053285326541
%N G.f. satisfies: A(x) = x + A( A(x)^4 - A(x)^10 ).
%C Compare to: G(x) = x + G( G(x)^4 - G(x)^16 ) holds when G(x) = x + G(x)^4 is a g.f. of A002293.
%C Compare to: F(x) = x + F( F(x)^3 - F(x)^9 ) holds when F(x) = x + F(x)^3 is a g.f. of the ternary tree numbers (A001764).
%H Paul D. Hanna, <a href="/A275758/b275758.txt">Table of n, a(n) for n = 1..300</a>
%F G.f. satisfies:
%F (1) A(x - A(x^4 - x^10)) = x.
%F (2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^4-x^10)^(n+1) / (n+1)!.
%F (3) A(x) = x * exp( Sum_{n>=0} d^n/dx^n A(x^4-x^10)^(n+1)/x / (n+1)! ).
%e G.f.: A(x) = x + x^4 + 4*x^7 + 21*x^10 + 126*x^13 + 817*x^16 + 5574*x^19 + 39418*x^22 + 286286*x^25 + 2122491*x^28 + 15995696*x^31 + 122166551*x^34 +...
%e such that A(x) = x + A( A(x)^4 - A(x)^10 ).
%e RELATED SERIES.
%e A(x)^4 = x^4 + 4*x^7 + 22*x^10 + 136*x^13 + 901*x^16 + 6248*x^19 + 44758*x^22 + 328520*x^25 + 2457286*x^28 + 18659736*x^31 + 143455026*x^34 +...
%e A(x)^10 = x^10 + 10*x^13 + 85*x^16 + 690*x^19 + 5520*x^22 + 44002*x^25 + 351045*x^28 + 2808040*x^31 + 22537355*x^34 + 181530280*x^37 + 1467320874*x^40 +...
%e A(x^4 - x^10) = x^4 - x^10 + x^16 - 4*x^22 + 10*x^28 - 32*x^34 + 106*x^40 - 350*x^46 + 1211*x^52 - 4242*x^58 + 15083*x^64 - 54404*x^70 + 198114*x^76 +...
%e where Series_Reversion(A(x)) = x - A(x^4 - x^10).
%o (PARI) {a(n) = my(A=x); for(i=1, 3*n, A = x + subst(A, x, A^4 - A^10 +x*O(x^(3*n)))); polcoeff(A, 3*n-2)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A275755, A275756, A275757.
%K nonn
%O 1,3
%A _Paul D. Hanna_, Aug 20 2016