A143139 E.g.f.: A(x) = exp(x + A(x)^2) - 1.

1, 3, 25, 351, 6901, 174483, 5392465, 196967991, 8301682141, 396555037803, 21171512707225, 1249311005445231, 80742309245690821, 5672134436846492163, 430345858647623635105, 35069095795843414698471, 3054896437732455928549741, 283283784773408059496473563

Offset

1,2

Comments

Conjecture: the sequence {a(n)} reducced modulo a positive integer k is eventually periodic with the period dividing phi(k), where phi(n) = A000010(n). For example, modulo 14 the sequence becomes [1, 3, 11, 1, 13, 1, 1, 3, 11, 1, 13, 1, 1, ...] with an apparent period of 6 = phi(14) starting at n = 2. - Peter Bala, Feb 18 2026

Links

Table of n, a(n) for n=1..18.

Peter Bala, The method of Graves for computing inverse functions.

Formula

E.g.f.: A(x) = Series_Reversion( log(1+x) - x^2 ).

E.g.f. derivative: A'(x) = (1 + A(x))/(1 - 2*A(x) - 2*A(x)^2 ).

a(n) = Sum_{k=0..n-1} (n+k-1)!*Sum_{j=0..k} (-1)^(j)/(k-j)!*Sum_{l=0..min(j,(n+j-1)/2)} (-1)^l*Stirling1(n-2*l+j-1,j-l)/(l!*(n-2*l+j-1)!). - Vladimir Kruchinin, Feb 17 2012

a(n) ~ sqrt(1+1/sqrt(3)) * 2^(n-3/2) * n^(n-1) / (exp(n) * (sqrt(3)-2-2*log(sqrt(3)-1))^(n-1/2)). - Vaclav Kotesovec, Dec 28 2013

a(n) = Sum_{k=0..n-1} Stirling2(n+k,2*k+1)*(2*k)!/k!. - Tani Akinari, Jan 06 2026

Example

A(x) = x + 3*x^2/2! + 25*x^3/3! + 351*x^4/4! + 6901*x^5/5! + ...
where A(log(1+x) - x^2) = x.
Log(1 + A(x)) = x + A(x)^2 = G(x) = g.f. of A143138:
G(x) = x + 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...
A(x)^2 = 2*x^2/2! + 18*x^3/3! + 254*x^4/4! + 5010*x^5/5! + ...

Maple

# uses Graves' method to calculate the coefficients in the k-th power of the e.g.f. A(x).
d := proc (n, x) option remember; if n = 0 then x^k else simplify( (1 + x)/(1 - 2*x*(1 + x)) * diff(  d(n-1, x), x) ) end if end proc:
k := 1:
seq( eval(d(n, x), x = 0), n = 0..20 ); # Peter Bala, Feb 14 2026

Mathematica

Rest[CoefficientList[InverseSeries[Series[Log[1+x]-x^2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Dec 28 2013 *)

Prog

(PARI) {a(n)=local(A=x+O(x^n)); for(i=0, n, A=exp(x+A^2)-1); n!*polcoeff(A, n)}
(PARI) {a(n)=n!*polcoeff(exp(serreverse(x-(exp(x+x*O(x^n))-1)^2))-1, n)}
(Maxima)
a(n):=sum((n+k-1)!*sum((-1)^(j)/(k-j)!*sum(((-1)^l*stirling1(n-2*l+j-1, j-l))/(l!*(n-2*l+j-1)!), l, 0, min(j, (n+j-1)/2)), j, 0, k), k, 0, n-1); /* Vladimir Kruchinin, Feb 17 2012 */
(Maxima) a(n):=sum(stirling2(n+k, 2*k+1)*(2*k)!/(k!), k, 0, n-1); /* Tani Akinari, Jan 06 2026 */

Crossrefs

Cf. A143138, A391954, A392723.

Sequence in context: A093360 A161629 A129506 * A231637 A394478 A377829

Adjacent sequences: A143136 A143137 A143138 * A143140 A143141 A143142

Keyword

nonn,easy

Author

Paul D. Hanna, Jul 27 2008

Status

approved