OFFSET
1,2
COMMENTS
Conjecture: a(n) is odd iff n is a power of 2 for n >= 1.
a(n) divisible by 3 for n > 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..500
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) A(x) = x + (x*A(x)^2)'.
(2) A(x) = x + A(x)^2 + 2*x*A(x)*A'(x).
(3) A(x) = (x + A(x)^2) / (1 - 2*x*A'(x)).
(4) A(x) = x + A(x) * (A(x) + 2*x*A'(x)).
(5) A(x) = x * exp( Integral (A(x) - x - 3*A(x)^2)/(2*x*A(x)^2) dx ).
a(n) = Sum_{k=1..n-1} (2*k+1) * a(k)*a(n-k) for n > 1 with a(1) = 1.
a(1) = 1; a(n) = (n+1) * Sum_{k=1..n-1} a(k)*a(n-k). - Seiichi Manyama, Jul 03 2026
a(n) ~ c * 2^n * n^(n+3) / exp(n), where c = 0.05703660816398865072657530036... - Vaclav Kotesovec, Jul 03 2026
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 24*x^3 + 285*x^4 + 4284*x^5 + 75978*x^6 + 1530720*x^7 + 34237485*x^8 + 837481140*x^9 + ...
where A(x) = x + (x*A(x)^2)'.
RELATED SERIES.
A(x)^2 = x^2 + 6*x^3 + 57*x^4 + 714*x^5 + 10854*x^6 + 191340*x^7 + 3804165*x^8 + 83748114*x^9 + ...
A'(x) = 1 + 6*x + 72*x^2 + 1140*x^3 + 21420*x^4 + 455868*x^5 + 10715040*x^6 + 273899880*x^7 + 7537330260*x^8 + ...
A(x)*A'(x) = x + 9*x^2 + 114*x^3 + 1785*x^4 + 32562*x^5 + 669690*x^6 + 15216660*x^7 + 376866513*x^8 + ...
where A(x) = x + A(x)^2 + 2*x*A(x)*A'(x).
PROG
(PARI) \\ by definition
{a(n) = my(A=x); for(i=1, n, A = x + deriv( x*A^2 ) +x*O(x^n) ); GF=A; polcoef(A, n)}
{upto(n) = a(n); Vec(GF)}
upto(20)
(PARI) \\ by recurrence (slow)
{a(n) = if(n==1, 1, sum(k=1, n-1, (2*k+1) * a(k)*a(n-k) ))}
for(n=1, 15, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jul 03 2026
STATUS
approved