A397588 - OEIS

%I #18 Jul 04 2026 10:54:05

%S 1,3,24,285,4284,75978,1530720,34237485,837481140,22170740526,

%T 630399084912,19141524214722,617861322312696,21122842522288500,

%U 762448141026794880,28980030721635717885,1157145229347334181604,48434042350706316375942,2121007271655295076291280,97002622108401462159296310

%N G.f. A(x) satisfies A(x) = x + (x*A(x)^2)'.

%C Conjecture: a(n) is odd iff n is a power of 2 for n >= 1.

%C a(n) divisible by 3 for n > 1.

%H Paul D. Hanna, <a href="/A397588/b397588.txt">Table of n, a(n) for n = 1..500</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) A(x) = x + (x*A(x)^2)'.

%F (2) A(x) = x + A(x)^2 + 2*x*A(x)*A'(x).

%F (3) A(x) = (x + A(x)^2) / (1 - 2*x*A'(x)).

%F (4) A(x) = x + A(x) * (A(x) + 2*x*A'(x)).

%F (5) A(x) = x * exp( Integral (A(x) - x - 3*A(x)^2)/(2*x*A(x)^2) dx ).

%F a(n) = Sum_{k=1..n-1} (2*k+1) * a(k)*a(n-k) for n > 1 with a(1) = 1.

%F a(1) = 1; a(n) = (n+1) * Sum_{k=1..n-1} a(k)*a(n-k). - _Seiichi Manyama_, Jul 03 2026

%F a(n) ~ c * 2^n * n^(n+3) / exp(n), where c = 0.05703660816398865072657530036... - _Vaclav Kotesovec_, Jul 03 2026

%e 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 + ...

%e where A(x) = x + (x*A(x)^2)'.

%e RELATED SERIES.

%e 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 + ...

%e 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 + ...

%e 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 + ...

%e where A(x) = x + A(x)^2 + 2*x*A(x)*A'(x).

%o (PARI) \\ by definition

%o {a(n) = my(A=x); for(i=1,n, A = x + deriv( x*A^2 ) +x*O(x^n) ); GF=A; polcoef(A,n)}

%o {upto(n) = a(n); Vec(GF)}

%o upto(20)

%o (PARI) \\ by recurrence (slow)

%o {a(n) = if(n==1,1, sum(k=1,n-1, (2*k+1) * a(k)*a(n-k) ))}

%o for(n=1,15, print1(a(n),", "))

%K nonn,new

%O 1,2

%A _Paul D. Hanna_, Jul 03 2026