A276358 G.f. A(x) satisfies: A(x - x*A(x)) = x + x*A(x).

1, 2, 8, 46, 324, 2608, 23136, 221370, 2252872, 24153284, 270922880, 3163154736, 38291322000, 479133266432, 6181998751808, 82084129578414, 1119798740473788, 15674024566862424, 224843628257016920, 3302256609111585300, 49613275311027132672, 761926428688868584400, 11952618573953745931536, 191418290850831848697272, 3127755564602007721663352, 52118116918762815035493760, 885205781290692080951844800, 15318116453244882343710519680, 269953482313408263924956600000

Offset: 1

Author

Paul D. Hanna, Sep 04 2016

Keywords

• nonn

Examples

			G.f.: A(x) = x + 2*x^2 + 8*x^3 + 46*x^4 + 324*x^5 + 2608*x^6 + 23136*x^7 + 221370*x^8 + 2252872*x^9 + 24153284*x^10 + 270922880*x^11 + 3163154736*x^12 +...
such that A(x - x*A(x)) = x + x*A(x).
RELATED SERIES.
A(x - x*A(x)) = x + x^2 + 2*x^3 + 8*x^4 + 46*x^5 + 324*x^6 + 2608*x^7 +...
which equals x + x*A(x).
Series_Reversion( x - x*A(x) ) = x + x^2 + 4*x^3 + 23*x^4 + 162*x^5 + 1304*x^6 + 11568*x^7 + 110685*x^8 + 1126436*x^9 + 12076642*x^10 + 135461440*x^11 + 1581577368*x^12 +...
which equals (x + A(x))/2.
A( (x + A(x))/2 ) = x + 3*x^2 + 16*x^3 + 111*x^4 + 898*x^5 + 8068*x^6 + 78400*x^7 + 810875*x^8 + 8832804*x^9 + 100592970*x^10 + 1191393144*x^11 + 14616198024*x^12 +...
which equals (A(x) - x)/(A(x) + x).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A275765.

Programs

PARI

{a(n) = my(A=x); for(i=1,n, A = 2*serreverse( x - x*A +x*O(x^n) ) - x ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

PARI

{a(n) = my(A=x, B); for(i=1,n, B = (x + A)/2 +x*O(x^n); A = x*(1 + subst(A,x,B))/(1 - subst(A,x,B)) ); polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A(x) = 2 * Series_Reversion( x - x*A(x) ) - x.

(2) A(x) = x * (1 + A(B(x))) / (1 - A(B(x))), where B(x) = (x + A(x))/2.

(3) A( (x + A(x))/2 ) = (A(x) - x) / (A(x) + x).