cp's OEIS Frontend

A276360 G.f. satisfies: A(x - A(x)^2) = x + 2*A(x)^2.

%I A276360 #12 Sep 30 2019 07:15:31
%S A276360 1,3,24,276,3858,61092,1056816,19550475,381543576,7782820548,
%T A276360 164842646424,3607654164924,81281990795520,1879865970374568,
%U A276360 44527769989124976,1078220967132218616,26650484274297181896,671558570413109457264,17234310756238557856200,450044549619831325213920,11949386806898017225833312,322394088574898542428753168,8833647058171126097908059720
%N A276360 G.f. satisfies: A(x - A(x)^2) = x + 2*A(x)^2.
%H A276360 Paul D. Hanna, <a href="/A276360/b276360.txt">Table of n, a(n) for n = 1..300</a>
%F A276360 G.f. A(x) also satisfies:
%F A276360 (1) A(x) = x + 3 * A( 2*x/3 + A(x)/3 )^2.
%F A276360 (2) A(x) = -2*x + 3 * Series_Reversion(x - A(x)^2).
%F A276360 (3) 2*R(x) = -x + 3 * Series_Reversion(x + 2*A(x)^2), where R(A(x)) = x.
%F A276360 (4) R( sqrt( x/3 - R(x)/3 ) ) = x/3 + 2*R(x)/3, where R(A(x)) = x.
%F A276360 a(n) = Sum_{k=0..n-1} A277295(n,k)*3^(n-k-1).
%e A276360 G.f.: A(x) = x + 3*x^2 + 24*x^3 + 276*x^4 + 3858*x^5 + 61092*x^6 + 1056816*x^7 + 19550475*x^8 + 381543576*x^9 + 7782820548*x^10 + 164842646424*x^11 + 3607654164924*x^12 +...
%e A276360 such that A(x - A(x)^2) = x + 2*A(x)^2.
%e A276360 RELATED SERIES.
%e A276360 Note that Series_Reversion(x - A(x)^2) = 2*x/3 + A(x)/3, which begins:
%e A276360 Series_Reversion(x - A(x)^2) = x + x^2 + 8*x^3 + 92*x^4 + 1286*x^5 + 20364*x^6 + 352272*x^7 + 6516825*x^8 + 127181192*x^9 + 2594273516*x^10 + 54947548808*x^11 +
%e A276360 1202551388308*x^12 +...
%e A276360 Let R(x) = Series_Reversion(A(x)) so that R(A(x)) = x,
%e A276360 R(x) = x - 3*x^2 - 6*x^3 - 51*x^4 - 564*x^5 - 7416*x^6 - 109764*x^7 - 1772028*x^8 - 30603930*x^9 - 558238326*x^10 - 10659285096*x^11 - 211688430204*x^12 +...
%e A276360 then Series_Reversion(x + 2*A(x)^2) = x/3 + 2*R(x)/3.
%t A276360 m = 24; A[_] = 0;
%t A276360 Do[A[x_] = x + 3 A[2 x/3 + A[x]/3]^2 + O[x]^m // Normal, {m}];
%t A276360 CoefficientList[A[x]/x, x] (* _Jean-François Alcover_, Sep 30 2019 *)
%o A276360 (PARI) {a(n) = my(A=[1],F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -polcoeff(subst(F, x, x-F^2) - 2*F^2, #A) ); A[n]}
%o A276360 for(n=1, 30, print1(a(n), ", "))
%Y A276360 Cf. A277295, A213591, A275765, A276361, A276362, A276363.
%K A276360 nonn
%O A276360 1,2
%A A276360 _Paul D. Hanna_, Aug 31 2016