A154757 G.f. satisfies A(x) = Sum_{n>=0} A^n(x)^(n+1).

1, 1, 3, 14, 83, 574, 4433, 37274, 335751, 3204616, 32159272, 337383121, 3684215369, 41735965645, 489173496890, 5919292080009, 73818803164143, 947369393139133, 12496581035619137, 169248547161053704, 2351389448608182950, 33484451978153816768, 488397582642492894240, 7291892040418905617996

Offset

1,3

Comments

Notation: A^n(x) denotes the n-th iteration of A(x) with A^0(x) = x.

Links

Paul D. Hanna, Table of n, a(n) for n = 1..150

Formula

G.f. A(x) satisfies: A^k(x) = Sum_{n>=0} ( A^(n+k-1)(x) )^(n+1) for all k.

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 14*x^4 + 83*x^5 + 574*x^6 + ...
where A(x) = x + A(x)^2 + A^2(x)^3 + A^3(x)^4 + A^4(x)^5 + ...
Likewise,
A^2(x) = A(x) + A^2(x)^2 + A^3(x)^3 + A^4(x)^4 + A^5(x)^5 + ...
A^3(x) = A^2(x) + A^3(x)^2 + A^4(x)^3 + A^5(x)^3 + A^6(x)^5 + ...
A^(-1)(x) = x - x^2 - A(x)^3 - A^2(x)^4 - A^3(x)^5 - ...
etc.
Explicitly,
A^2(x) = x + 2*x^2 + 8*x^3 + 44*x^4 + 292*x^5 + 2201*x^6 + ...;
A^3(x) = x + 3*x^2 + 15*x^3 + 96*x^4 + 715*x^5 + 5921*x^6 + ...;
A^4(x) = x + 4*x^2 + 24*x^3 + 176*x^4 + 1464*x^5 + 13322*x^6 + ...;
A^(-1)(x) = x - x^2 - x^3 - 4*x^4 - 21*x^5 - 133*x^6 - 959*x^7 - ...;
also,
A(x)^2 = x^2 + 2*x^3 + 7*x^4 + 34*x^5 + 203*x^6 + 1398*x^7 + ...;
A^2(x)^3 = x^3 + 6*x^4 + 36*x^5 + 236*x^6 + 1692*x^7 + ...;
A^3(x)^4 = x^4 + 12*x^5 + 114*x^6 + 1032*x^7 + 9367*x^8 + ...;
A^4(x)^5 = x^5 + 20*x^6 + 280*x^7 + 3440*x^8 + ...;
A^5(x)^6 = x^6 + 30*x^7 + 585*x^8 + 9490*x^9 + ...; ...

Prog

(PARI) \\ By definition A(x) = Sum_{n>=0} A^n(x)^(n+1)
{a(n) = my(A=x, G); for(k=1, n, A = truncate(A) + x*O(x^k); G = x + x*O(x^k);
A = x + sum(m=1, k, (G=subst(A, x, G))^(m+1)) + x*O(x^n); ); polcoeff(GF=A, n)}
\\ Print the terms
{upto(n) = a(n); Vec(GF)}
upto(25) \\ Program revised by Paul D. Hanna, Jun 10 2026

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

Extensions

Entry revised by Paul D. Hanna, Jun 10 2026

Status

approved