OFFSET
0,2
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 has the following properties.
(1) 3^(n-1) = Sum_{k=0..n-1} [x^k] A(x/n)^n for n >= 1.
(2) (3*n)^(n-1) = [x^(n-1)] A(x)^n / (1 - n*x) for n >= 1.
(3) (3*n)^(n-1) = [x^(n-1)] B'(x) / (1 - n*B(x)) for n >= 1 where B(x) = x*A(B(x)) and B(x) = Series_Reversion(x/A(x)).
(4) [x^(n-1)] B'(x)/(1 - B(x)) = Sum_{k=0..n-1} [x^k] A(x)^n for n >= 1 where B(x) = x*A(B(x)).
a(n) = Sum_{k=0..n} 3^k * A397590(n,k) for n >= 0.
a(n) ~ (exp(1) - 1/3) * exp(-2*exp(-1)/3) * 3^n * n^(n-1). - Vaclav Kotesovec, Jul 11 2026
EXAMPLE
G.f.: A(x) = 1 + 2*x + 14*x^2 + 212*x^3 + 5060*x^4 + 163612*x^5 + 6627896*x^6 + 321431160*x^7 + 18124740188*x^8 + ...
where 3^(n-1) = Sum_{k=0..n-1} [x^k] A(x/n)^n for n >= 1, as illustrated below.
The table of coefficients of x^k in A(x)^n begins
n = 1: [1, 2, 14, 212, 5060, 163612, ...];
n = 2: [1, 4, 32, 480, 11164, 353400, ...];
n = 3: [1, 6, 54, 812, 18480, 573084, ...];
n = 4: [1, 8, 80, 1216, 27192, 826832, ...];
n = 5: [1, 10, 110, 1700, 37500, 1119292, ...];
n = 6: [1, 12, 144, 2272, 49620, 1455624, ...]; ...
The sums of the first n coefficients in A(x/n)^n are
n = 1: 3^0 = 1;
n = 2: 3^1 = 1 + 4/2;
n = 3: 3^2 = 1 + 6/3 + 54/3^2;
n = 4: 3^3 = 1 + 8/4 + 80/4^2 + 1216/4^3;
n = 5: 3^4 = 1 + 10/5 + 110/5^2 + 1700/5^3 + 37500/5^4;
n = 6: 3^5 = 1 + 12/6 + 144/6^2 + 2272/6^3 + 49620/6^4 + 1455624/6^5;
etc.
RELATED SERIES.
Let B(x) = Series_Reversion(x/A(x)) then
B(x) = x + 2*x^2 + 18*x^3 + 304*x^4 + 7500*x^5 + 242604*x^6 + 9707376*x^7 + 463312384*x^8 + ...
where B(x) = x*A(B(x)) and
B'(x)/(1 - B(x)) = 1 + 5*x + 61*x^2 + 1305*x^3 + 39321*x^4 + 1507673*x^5 + 69860085*x^6 + ...
the coefficients of which are partial sums of [x^k] A(x)^n:
1 = 1, 5 = (1 + 4), 61 = (1 + 6 + 54), 1305 = (1 + 8 + 80 + 1216), ...
The table of coefficients in B'(x)/(1 - n*B(x)) begins
n = 1: [1, 5, 61, 1305, 39321, ...];
n = 2: [1, 6, 70, 1416, 41436, ...];
n = 3: [1, 7, 81, 1555, 43941, ...];
n = 4: [1, 8, 94, 1728, 46956, ...];
n = 5: [1, 9, 109, 1941, 50625, ...]; ...
in which the main diagonal equals
[1, 6, 81, 1728, 50625, ..., (3*n)^(n-1), ...].
PROG
(PARI) \\ by definition
{a(n, y=3) = my(A=[1], m, V); for(i=1, n, A = concat(A, 0); m=#A; V=Vec( subst(Ser(A)^m, x, x/m) );
A[m] = (y^(m-1) - sum(k=1, #V, V[k]) )*m^(m-2) ); GF=Ser(A); polcoef(GF, n)}
{upto(n, y=3) = a(n, y); Vec(GF)}
upto(25)
(PARI) \\ from (3*n)^(n-1) = [x^(n-1)] A(x)^n / (1 - n*x)
{a(n, y=3) = my(A=[1], m, V); for(i=1, n, A = concat(A, 0); m=#A;
A[m] = ((y*m)^(m-1) - polcoef(Ser(A)^m/(1-m*x), m-1))/m ); GF=Ser(A); polcoef(GF, n)}
{upto(n, y=3) = a(n, y); Vec(GF)}
upto(25)
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jul 10 2026
STATUS
approved