A154757 G.f. A(x) = Sum_{n>=0} A_n(x)^(n+1) where A_{n+1}(x) = A_n( A(x) ) with A_0(x)=x and A(0)=0.

1, 1, 3, 14, 83, 574, 4433, 37274, 335751, 3204616, 32159272, 337383121, 3684215369, 41735965645, 489173496890, 5919292080009, 73818803114991, 947369389903293, 12496580900037441, 169248542357479944

Offset

1,3

Links

Table of n, a(n) for n=1..20.

Formula

G.f. A(x) satisfies: A_n(x) = Sum_{k>=0} ( A_{n+k-1}(x) )^(k+1) ; e.g.:

A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 + A(A(A(A(x))))^5 + ...;

A(A(x)) = A(x) + A(A(x))^2 + A(A(A(x)))^3 + A(A(A(A(x))))^4 + ...;

A(A(A(x))) = A(A(x)) + A(A(A(x)))^2 + A(A(A(A(x))))^3 + ...; ...

Series_Reversion(A(x)) = x - x^2 - A(x)^3 - A(A(x))^4 - A(A(A(x)))^5 - ...

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 83*x^5 + 574*x^6 + ...;
A(A(x)) = x + 2*x^2 + 8*x^3 + 44*x^4 + 292*x^5 + 2201*x^6 + ...;
A(A(A(x))) = x + 3*x^2 + 15*x^3 + 96*x^4 + 715*x^5 + 5921*x^6 + ...;
A(A(A(A(x)))) = x + 4*x^2 + 24*x^3 + 176*x^4 + 1464*x^5 + 13322*x^6 + ...; ...
Series_Reversion(A(x)) = x - x^2 - x^3 - 4*x^4 - 21*x^5 - 133*x^6 - 959*x^7 - ...
ILLUSTRATE DEFINITION:
A(x) = x + A(x)^2 + A(A(x))^3 + A(A(A(x)))^4 + ... where:
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...;
A(A(x))^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1692*x^7 + ...;
A(A(A(x)))^4 = x^4 + 12*x^5 + 114*x^6 + 1032*x^7 + 9367*x^8 + ...;
A(A(A(A(x))))^5 = x^5 + 20*x^6 + 280*x^7 + 3440*x^8 + ...;
A(A(A(A(A(x)))))^6 = x^6 + 30*x^7 + 585*x^8 + 9490*x^9 + ...; ...

Prog

(PARI) {a(n)=local(A=x, G, S); if(n<1, 0, for(j=1, n, G=x; S=x; for(i=2, n+1, G=subst(A, x, G+x*O(x^n)); S=S+G^i); A=S); polcoeff(A+x*O(x^n), n))}

Crossrefs

Sequence in context: A224776 A220910 A121687 * A352307 A359335 A074535

Adjacent sequences: A154754 A154755 A154756 * A154758 A154759 A154760

Keyword

nonn

Author

Paul D. Hanna, Jan 15 2009

Status

approved