A206293 G.f. satisfies: A(x) = Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/A(x^k) ).

1, 1, 2, 5, 18, 78, 415, 2467, 16212, 114623, 863229, 6858780, 57156213, 497147291, 4497291265, 42189445764, 409478828567, 4103901097024, 42403116824997, 451059832858894, 4933844398096693, 55436157047213427, 639215949145395559, 7557505365363885063

Offset: 0

Author

Paul D. Hanna, Feb 05 2012

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 78*x^5 + 415*x^6 + 2467*x^7 +...
such that, by definition,
A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...
where G_n(x) satisfies: G_n( x/A(x^n) ) = x.
The first few expansions of G_n(x) begin:
G_1(x) = x + x^2 + 3*x^3 + 12*x^4 + 59*x^5 + 329*x^6 + 2035*x^7 +...
G_2(x) = x + x^3 + 4*x^5 + 22*x^7 + 144*x^9 + 1045*x^11 + 8159*x^13 +...
G_3(x) = x + x^4 + 5*x^7 + 35*x^10 + 289*x^13 + 2626*x^16 +...
G_4(x) = x + x^5 + 6*x^9 + 51*x^13 + 510*x^17 + 5597*x^21 +...
G_5(x) = x + x^6 + 7*x^11 + 70*x^16 + 823*x^21 + 10608*x^26 +...
G_6(x) = x + x^7 + 8*x^13 + 92*x^19 + 1244*x^25 + 18434*x^31 +...
G_7(x) = x + x^8 + 9*x^15 + 117*x^22 + 1789*x^29 + 29975*x^36 +...
where G_n(x) = x*A( G_n(x)^n ).
		

Crossrefs

Cf. A206290.

Programs

PARI

{a(n)=local(A=1+x);for(i=1,n,A=sum(m=0,n,prod(k=1,m,serreverse(x/subst(A,x,x^k +x*O(x^n))))));polcoeff(A,n)}
for(n=0,45,print1(a(n),", "))

Formula

G.f.: A(x) = Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:

(1) G_n(x) = Series_Reversion( x/A(x^n) ),

(2) G_n(x) = x * A( G_n(x)^n ).