A271960 G.f. A(x) satisfies: A(x)^2 = A( (x + 2*A(x)^2)^2 ).
1, 2, 9, 50, 312, 2086, 14613, 105864, 786627, 5962110, 45914544, 358247214, 2825957294, 22499804332, 180573770279, 1459277489372, 11864714598122, 96985441764430, 796580710229999, 6570692234061404, 54408498662798180, 452104483291381134, 3768693666865385520, 31506775300298343840, 264103426399414754616, 2219265880819182687882, 18690831189839369444283, 157746446435747834724764, 1333944139058301773582424
Offset: 1
Keywords
Examples
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 +... where A(x)^2 = A( (x + 2*A(x)^2)^2 ). RELATED SERIES. 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 +... (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 +... 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 +... Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then 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) +... such that B(x) = x - 2*x^2 - x*G(x^2), where G(x) = x + (1/2)*(G(x)^2 + G(x^2)). Let C(x) = series reversion of sqrt(x*A(x)), so that C(x)*A(C(x)) = x^2, then 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 +... where F(x)^2 = F( x^2 + 2*F(x)^3 ). SPECIFIC VALUES. A(1/9) = 0.193716899378316704135816541113258730084910624425207... where A(1/9)^2 = A( (1/9 + 2*A(1/9)^2)^2 ). A(1/10) = 0.14546429517891287614768130046564115776024954490631... where A(1/10)^2 = A( (1/10 + 2*A(1/10)^2)^2 ). A(1/11) = 0.12312262826542819891757177602341377693863412904549... where A(1/11)^2 = A( (1/11 + 2*A(1/11)^2)^2 ). A(1/12) = 0.10787967285688466984977060963500355819842649026285... where A(1/12)^2 = A( (1/12 + 2*A(1/12)^2)^2 ).
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..400
Programs
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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)} for(n=1, 30, print1(a(n), ", "))
Formula
G.f. A(x) satisfies:
(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).
(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.
a(n) ~ c * d^n / n^(3/2), where d = 8.9175668047902516038346068989... and c = 0.056993950617012713508863076... . - Vaclav Kotesovec, May 03 2016