OFFSET
1,3
COMMENTS
Essentially the same as column 0 in triangle A397590 (apart from signs and offset by 1).
Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..400
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n has the following properties.
(1) [x^(n-1)] (1 - A(x))^n / (1 - n*x) = 0 for n > 1.
(2) Sum_{k=0..n-1} [x^k] (1 - A(x/n))^n = 0 for n > 1.
(3) Sum_{k=0..n-1} n^(n-k) * [x^k] (1 - A(x))^n = 0 for n > 1.
(4) Sum_{k=0..n-1} n^(n-k) * Sum_{j=0..n} (-1)^j * binomial(n,j) * [x^k] A(x)^j = 0 for n > 1.
(5) 1/(1 - A(x)) equals the g.f. of A320415.
(6) a(n) = -A397590(n,0) for n >= 1.
a(n) ~ (exp(1) - 1) * exp(exp(-1) - 1) * n^(n-1). - Vaclav Kotesovec, Jul 11 2026
EXAMPLE
G.f.: A(x) = x + x^2 + 4*x^3 + 28*x^4 + 281*x^5 + 3658*x^6 + 58257*x^7 + 1093228*x^8 + 23558599*x^9 + 572266776*x^10 + ...
The table of coefficients of x^k in (1 - A(x))^n begins
n = 1: [1, -1, -1, -4, -28, -281, -3658, ...];
n = 2: [1, -2, -1, -6, -47, -498, -6682, ...];
n = 3: [1, -3, 0, -7, -60, -666, -9181, ...];
n = 4: [1, -4, 2, -8, -69, -796, -11242, ...];
n = 5: [1, -5, 5, -10, -75, -896, -12935, ...];
n = 6: [1, -6, 9, -14, -78, -972, -14317, ...];
n = 7: [1, -7, 14, -21, -77, -1029, -15435, ...];
...
from which we may illustrate that the sum of the first n coefficients in (1 - A(x/n))^n equals 0 for n > 1 by:
n = 1: 1 = 1;
n = 2: 0 = 1 - 2/2;
n = 3: 0 = 1 - 3/3 + 0/3^2;
n = 4: 0 = 1 - 4/4 + 2/4^2 - 8/4^3;
n = 5: 0 = 1 - 5/5 + 5/5^2 - 10/5^3 - 75/5^4;
n = 6: 0 = 1 - 6/6 + 9/6^2 - 14/6^3 - 78/6^4 - 972/6^5;
etc.
RELATED SERIES.
Given g.f. A(x), then 1/(1 - A(x)) is the g.f. of A320415
1/(1 - A(x)) = 1 + x + 2*x^2 + 7*x^3 + 41*x^4 + 365*x^5 + 4429*x^6 + 67631*x^7 + 1235436*x^8 + ... + A320415(n)*x^n + ...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=2, n, A=concat(A, 0); m=#A; A[m] = Vec( (1-Ser(A))^m / (1 - m*x +x^2*O(x^m)) )[m]/m ); GF=Ser(A); A[n+1]}
{upto(n) = a(n); Vec(GF)}
upto(25)
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jul 09 2026
STATUS
approved