OFFSET
0,2
COMMENTS
At the same time that I introduced the polynomials P(n,x) defined by P(0,x)=1 and for n>0, P(n,x) = (-1)^n/(n+1) + x*Sum_{ i=0..n-1 } ( (-1)^i/(i+1) )*P(n-1-i,x) (Gazette des Mathematiciens 1992), I gave the generalization P(0,x) = u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } u(i)*P(n-1-i,x).
For u(n), n>=0, = 1 1 1 2 3 4 5 6 7 8 ... the array of coefficients of the polynomials P(n,x) is:
1
1 1
1 2 1
2 3 3 1
3 6 6 4 1
4 11 13 10 5 1
5 18 27 24 15 6 1
6 28 51 55 40 21 7 1
whose row sums are the present sequence.
The alternating row sums are 1 0 0 1 0 0 0 -1 ...
The antidiagonal sums are 1 1 2 4 7 13 23 41 73 ...
The first column of the inverse matrix is 1 -1 1 -2 5 -11 25 -63 ...
REFERENCES
Paul Curtz, Gazette des Mathématiciens, 1992, no. 52, p. 44.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: (1-x+x^3)/(1-3*x+2*x^2-x^4). - Alois P. Heinz, Oct 14 2009
MAPLE
a:= n-> (Matrix([1, 1, 0, 1]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1][i] else 0 fi)^n)[1, 1]:
seq(a(n), n=0..50); # Alois P. Heinz, Oct 14 2009
MATHEMATICA
u[n_ /; n < 3] = 1; u[n_] := n-1;
p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[ u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand;
row[n_] := CoefficientList[ p[n][x], x];
Table[row[n] // Total, {n, 0, 30}] (* Jean-François Alcover, Oct 02 2012 *)
PROG
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^3)/(1-3*x+2*x^2-x^4) )); // G. C. Greubel, Oct 24 2023
(SageMath)
def A129891_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-x+x^3)/(1-3*x+2*x^2-x^4) ).list()
A129891_list(40) # G. C. Greubel, Oct 24 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul Curtz, Jun 04 2007
EXTENSIONS
Edited by N. J. A. Sloane, Jul 05 2007
More terms from Alois P. Heinz, Oct 14 2009
STATUS
approved