0,2

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 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 ...

P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

G.f.: -(x^3-x+1)/(x^4-2*x^2+3*x-1). [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009]

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); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009]

Sums of coefficients of polynomials defined in A140530.

Cf. A129841, A129696, A130620.

Sequence in context: A123720 A034007 A109975 this_sequence A130587 A129988 A035530

Adjacent sequences: A129888 A129889 A129890 this_sequence A129892 A129893 A129894

nonn

Paul Curtz (bpcrtz(AT)free.fr), Jun 04 2007

Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 05 2007

More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009