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A271959 G.f. A(x) satisfies: A(x)^2 = A( x^2 + 2*A(x)^3 ), with A(0)=0, A'(0)=1.

%I A271959 #20 Aug 10 2024 03:54:49
%S A271959 1,1,3,11,46,206,968,4706,23475,119473,617911,3238299,17159235,
%T A271959 91778276,494844531,2686731634,14676812392,80608719674,444853616294,
%U A271959 2465582096210,13718412623437,76596933043436,429046752044241,2410260414325754,13576401507921773,76660997896449976,433862051647272420,2460614616704514931,13982516582761530427,79601022639916142384
%N A271959 G.f. A(x) satisfies: A(x)^2 = A( x^2 + 2*A(x)^3 ), with A(0)=0, A'(0)=1.
%C A271959 Compare the g.f. to the related identity:
%C A271959 C(x)^2 = C( x^2 - 2*C(x)^3 ), where C(x) = x - C(x)^2.
%H A271959 Paul D. Hanna, <a href="/A271959/b271959.txt">Table of n, a(n) for n = 1..300</a>
%F A271959 G.f. A(x) satisfies: A(x - x*G(x)) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of the Wedderburn-Etherington numbers (A001190).
%F A271959 G.f. A(x) satisfies: A( sqrt(x*F(x)) ) = F(x), where F(x) is the g.f. of A271960 and F(x)^2 = F( (x + 2*F(x)^2)^2 ). - _Paul D. Hanna_, Aug 09 2024
%F A271959 a(n) ~ c * d^n / n^(3/2), where d = 5.99301788836820936729... and c = 0.060358293047581601577... . - _Vaclav Kotesovec_, May 03 2016
%e A271959 G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 206*x^6 + 968*x^7 + 4706*x^8 + 23475*x^9 + 119473*x^10 + 617911*x^11 + 3238299*x^12 +...
%e A271959 where A(x)^2 = A( x^2 + 2*A(x)^3 ).
%e A271959 RELATED SERIES.
%e A271959 A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 28*x^5 + 123*x^6 + 570*x^7 + 2745*x^8 + 13596*x^9 + 68818*x^10 + 354380*x^11 + 1850642*x^12 + 9777476*x^13 + 52166536*x^14 +...
%e A271959 A(x)^3 = x^3 + 3*x^4 + 12*x^5 + 52*x^6 + 240*x^7 + 1152*x^8 + 5694*x^9 + 28776*x^10 + 148008*x^11 + 772208*x^12 + 4076736*x^13 + 21737472*x^14 +...
%e A271959 Let B(x) be the series reversion of g.f. A(x), A(B(x)) = x, then
%e A271959 B(x) = x - x^2 - x^3 - x^4 - 2*x^5 - 3*x^6 - 6*x^7 - 11*x^8 - 23*x^9 - 46*x^10 - 98*x^11 - 207*x^12 - 451*x^13 - 983*x^14 +...+ -A001190(n)*x^(n+1) +...
%e A271959 such that B(x) = x - x*G(x), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)).
%e A271959 SPECIFIC VALUES.
%e A271959 A(1/6) = 0.268639354433758631443638721883026384052966634356654...
%e A271959 where A(1/6)^2 = A( 1/36 + 2*A(1/6)^3 ).
%e A271959 A(1/7) = 0.185430467497916613031797200968643881842140126550450...
%e A271959 where A(1/7)^2 = A( 1/49 + 2*A(1/7)^3 ).
%e A271959 A(1/8) = 0.152603770337160474296825145654940422398214899394916...
%e A271959 where A(1/8)^2 = A( 1/64 + 2*A(1/8)^3 ).
%e A271959 A(1/10) = 0.11495842465953100301539082058016718103413837897733...
%e A271959 where A(1/10)^2 = A( 1/100 + 2*A(1/10)^3 ).
%o A271959 (PARI) {a(n) = my(A=x+x^2,X=x+x*O(x^n)); for(i=1,n, A = subst(A,x, X^2 + 2*A^3)^(1/2) ); polcoeff(A,n)}
%o A271959 for(n=1,30,print1(a(n),", "))
%Y A271959 Cf. A001190, A271960.
%K A271959 nonn
%O A271959 1,3
%A A271959 _Paul D. Hanna_, Apr 18 2016