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

0,2

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

Sums of coefficients of polynomials defined in A140530.

Cf. A129841, A129696, A130620.

Sequence in context: A266930 A034007 A109975 * A130587 A129988 A035530

Adjacent sequences: A129888 A129889 A129890 * A129892 A129893 A129894

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