OFFSET
0,3
COMMENTS
Conjecture: a(n) == 1 (mod 3) iff n+2 is of the form 3^m or 2*3^m.
Conjecture: a(n) == 2 (mod 3) iff n+2 is a sum of 2 distinct powers of 3 (A038464).
The limit lim_{n->oo} a(n)/(4^n * n!^2) appears to be finite and nonzero. At n = 900, a(900)/(4^900 * 900!^2) = 0.218727822653...
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..400
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = exp( A(x) - 1 - Sum_{n>=2} a(n)*x^n / (4*n^2) ).
(2) A(x) = B(x*A(x)^2) where B(x) = A(x/B(x)^2) is the g.f. of A397244 which satisfies (2*n)*[x^n] B(x)^(2*n) = (2*n-1)*[x^n] B(x)^(2*n+1) for n > 1.
(3) A'(x) = A(x) * (1 + Sum_{n>=1} (2*n+1)*(2*n+3)/(4*(n+1)) * a(n+1) * x^n ).
(4) a(n) = 4*n^2/(4*n^2-1) * [x^n] log(A(x)) for n > 1.
(5) a(n) = n*A396846(n)/(n^2-1) for n > 1, where A396846 is the logarithmic derivative of this sequence.
(6) a(n) = 4*n*a(n-1) + n*Sum_{k=1..n-2} (2*k+1)*(2*k+3)/(k+1) * a(k+1)*a(n-k-1) for n > 1, with a(0) = a(1) = 1.
(7) A(x) = 1+x + 4*x*(A(x)-1)*(A'(x) + x*A''(x))/A(x) + 4*x^2*A'(x)^2/A(x)^2. - Paul D. Hanna, Jun 26 2026
EXAMPLE
G.f.: A(x) = 1 + x + 8*x^2 + 276*x^3 + 19216*x^4 + 2109180*x^5 + 327968280*x^6 + 68141519908*x^7 + 18217107336224*x^8 + ...
where
A(x) = exp( x + 15*8*x^2/(4*2^2) + 35*276*x^3/(4*3^2) + 63*19216*x^4/(4*4^2) + 99*2109180*x^5/(4*5^2) + 143*327968280*x^6/(4*6^2) + ... + (4*n^2-1)*a(n)*x^n/(4*n^2) + ... ).
RELATED SERIES.
The logarithm of the g.f. A(x) begins
log(A(x)) = x + 15*x^2/2 + 805*x^3/3 + 75663*x^4/4 + 10440441*x^5/5 + 1954144335*x^6/6 + 474557013645*x^7/7 + ... + A396846(n)*x^n/n + ...
where A396846(n) = (4*n^2-1)*a(n)/(4*n) for n > 1.
Let B(x) be the g.f. of A397241, which begins
B(x) = 1 + x + 6*x^2 + 235*x^3 + 17010*x^4 + 1912023*x^5 + 302087720*x^6 + 63489224916*x^7 + 17121352485210*x^8 + ... + A397244(n)*x^n + ...
then A(x) = B(x*A(x)^2) and B(x) = A(x/B(x)^2).
The terms a(n) are divisible by n for n > 0:
a(n)/n = [1, 4, 92, 4804, 421836, 54661380, 9734502844, ...].
PROG
(PARI) \\ by a(n) = 4*n^2/(4*n^2-1) * [x^n] log(A) for n > 1:
{a(n) = my(V=[1, 1], A, m); for(i=2, n, V=concat(V, 0); A = Ser(V); m = #V-1;
V[#V] = (4*m^2-1)*polcoef(A, m) - 4*m^2*polcoef(log(A), m) ); GF=Ser(V); V[n+1]}
\\ returns a vector of terms a(k) for k = 0..n
{upto(n) = a(n); Vec(GF)}
upto(20)
(PARI) \\ by recurrence (slower)
\\ print terms a(n) for n = 0..N
N = 20; A = vector(N+1);
{a(n) = A[n+1] = if(n<=1, 1, 4*n*A[n] + n*sum(k=1, n-2, (2*k+1)*(2*k+3)/(k+1) * a(k+1) * A[n-k]) )}
for(n=0, N, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 24 2026
STATUS
approved