|
%I #13 May 03 2016 05:38:11
%S 1,2,9,50,312,2086,14613,105864,786627,5962110,45914544,358247214,
%T 2825957294,22499804332,180573770279,1459277489372,11864714598122,
%U 96985441764430,796580710229999,6570692234061404,54408498662798180,452104483291381134,3768693666865385520,31506775300298343840,264103426399414754616,2219265880819182687882,18690831189839369444283,157746446435747834724764,1333944139058301773582424
%N G.f. A(x) satisfies: A(x)^2 = A( (x + 2*A(x)^2)^2 ).
%H Paul D. Hanna, <a href="/A271960/b271960.txt">Table of n, a(n) for n = 1..200</a>
%F G.f. A(x) satisfies:
%F (1) A(x - 2*x^2 - x*G(x^2)) = x, where G(x) = x + (1/2)*(G(x)^2 + G(x^2)) is the g.f. of the Wedderburn-Etherington numbers (A001190).
%F (2) A(x) = F( sqrt(x*A(x)) ) where F(x)^2 = F( x^2 + 2*F(x)^3 ) and F(x) is the g.f. of A271959.
%F a(n) ~ c * d^n / n^(3/2), where d = 8.9175668047902516038346068989... and c = 0.056993950617012713508863076... . - _Vaclav Kotesovec_, May 03 2016
%e G.f.: A(x) = x + 2*x^2 + 9*x^3 + 50*x^4 + 312*x^5 + 2086*x^6 + 14613*x^7 + 105864*x^8 + 786627*x^9 + 5962110*x^10 + 45914544*x^11 + 358247214*x^12 +...
%e where A(x)^2 = A( (x + 2*A(x)^2)^2 ).
%e RELATED SERIES.
%e A(x)^2 = x^2 + 4*x^3 + 22*x^4 + 136*x^5 + 905*x^6 + 6320*x^7 + 45686*x^8 + 338928*x^9 + 2565688*x^10 + 19739244*x^11 + 153893122*x^12 +...
%e (x + 2*A(x)^2)^2 = x^2 + 4*x^3 + 20*x^4 + 120*x^5 + 784*x^6 + 5412*x^7 + 38808*x^8 + 286200*x^9 + 2156704*x^10 + 16533088*x^11 + 128521172*x^12 +...
%e sqrt(x*A(x)) = x + x^2 + 4*x^3 + 21*x^4 + 127*x^5 + 832*x^6 + 5746*x^7 + 41191*x^8 + 303602*x^9 + 2286359*x^10 + 17515640*x^11 + 136074960*x^12 + 1069490964*x^13 + 8488634979*x^14 + 67943128844*x^15 + 547784144486*x^16 +...
%e Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
%e B(x) = x - 2*x^2 - x^3 - x^5 - x^7 - 2*x^9 - 3*x^11 - 6*x^13 - 11*x^15 - 23*x^17 - 46*x^19 - 98*x^21 - 207*x^23 - 451*x^25 +...+ -A001190(n)*x^(2*n+1) +...
%e such that B(x) = x - 2*x^2 - x*G(x^2), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)).
%e Let C(x) = series reversion of sqrt(x*A(x)), so that C(x)*A(C(x)) = x^2, then
%e A(C(x)) = F(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 +...+ A271959(n)*x^n +...
%e where F(x)^2 = F( x^2 + 2*F(x)^3 ).
%o (PARI) {a(n) = my(A=x+x^2, X=x+x*O(x^n)); for(i=1, n, A = subst(A, x, (X + 2*A^2)^2 )^(1/2) ); polcoeff(A, n)}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A271959, A001190.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Apr 23 2016
| 