A271844 G.f. A(x) satisfies: A(x) = x + A( A(x)^2 + A(x)^4 ).

1, 1, 2, 7, 26, 102, 420, 1793, 7854, 35106, 159492, 734334, 3418892, 16068532, 76135112, 363283763, 1744135306, 8419281306, 40838500796, 198950342814, 972999755364, 4775441138580, 23513016382120, 116111875760294, 574927064750460, 2853800953323468, 14197997592237912, 70786396399962476, 353611516341840008, 1769694222850151128

Offset: 1

Author

Paul D. Hanna, Apr 15 2016

Keywords

• nonn

Comments

Compare g.f. to: C(x) = x + C( C(x)^2 - C(x)^4 ) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 26*x^5 + 102*x^6 + 420*x^7 + 1793*x^8 + 7854*x^9 + 35106*x^10 + 159492*x^11 + 734334*x^12 +...
where A(x) = x + A( A(x)^2 + A(x)^4 ).
RELATED SERIES.
A(x)^2 + A(x)^4 = x^2 + 2*x^3 + 6*x^4 + 22*x^5 + 84*x^6 + 340*x^7 + 1434*x^8 + 6226*x^9 + 27632*x^10 + 124820*x^11 + 572000*x^12 +...
A(x^2 + x^4) = x^2 + 2*x^4 + 4*x^6 + 14*x^8 + 60*x^10 + 276*x^12 + 1320*x^14 + 6530*x^16 + 33188*x^18 + 172252*x^20 + 909016*x^22 +...
where the series reversion of A(x) equals x - A(x^2 + x^4).
		

Links

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

Crossrefs

Cf. A155134, A141200.

Programs

PARI

{a(n) = my(A=x+x^2 +x*O(x^n)); for(i=1,n, A = x + subst(A,x,A^2 + A^4) ) ; polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) A( x - A(x^2 + x^4) ) = x.

(2) A(x) = x + Sum_{n>=0} d^n/dx^n A(x^2+x^4)^(n+1) / (n+1)!.

(3) A(x) = x*exp( Sum_{n>=0} d^n/dx^n A(x^2+x^4)^(n+1)/x / (n+1)! ).

a(n) ~ c * d^n / n^(3/2), where d = 5.26908951612012208739853420341892... and c = 0.063610446185354820395355587671... . - Vaclav Kotesovec, Apr 16 2016