A129891 Sum of coefficients of polynomials defined in comments lines.

1, 2, 4, 9, 20, 44, 96, 209, 455, 991, 2159, 4704, 10249, 22330, 48651, 105997, 230938, 503150, 1096225, 2388372, 5203604, 11337218, 24700671, 53815949, 117250109, 255455647, 556567394, 1212606837, 2641935832, 5756049469, 12540844137

Offset: 0

Author

Paul Curtz, Jun 04 2007

Keywords

• nonn

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

Crossrefs

Sums of coefficients of polynomials defined in A140530.

Cf. A129841, A129696, A130620.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^3)/(1-3*x+2*x^2-x^4) )); // G. C. Greubel, Oct 24 2023

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 *)

SageMath

def A129891_list(prec):
    P. = 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

Formula

G.f.: (1-x+x^3)/(1-3*x+2*x^2-x^4). - Alois P. Heinz, Oct 14 2009

Extensions

Edited by N. J. A. Sloane, Jul 05 2007

More terms from Alois P. Heinz, Oct 14 2009