0,1

Given any sequence {u(i), i >= 0} we define a family of polynomials by P(0,x) = u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } (u(i)*P(n-i-1, x).

Then a(n) is the sum of the odd coefficients of P(n,x) if n is odd and a(n) is the sum of the even coefficients otherwise: a(n) = ((-1)^n*P(n,-1) +P(n,1))/2.

For the present exanmple we take {u(i)} to be 3,1,4,1,5,9,... (A000796).

P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.

P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .

We have P(0,x)=3, P(1,x)=1+9x, P(2,x)=4+6x+27x^2, ..., so that for example a(2) = (25+37)/2 = 31.

The polynomials P(n,x) are:

n=0: 3

n=1: 1+ 9*x

n=2: 4+ 6*x+ 27*x^2

n=3: 1+25*x+ 27*x^2+ 81*x^3

n=4: 5+14*x+117*x^2+108*x^3+243*x^4

n=5: 9+48*x+100*x^2+486*x^3+405*x^4+729*x^5

u:= proc(n) Digits:= max(n+10);

trunc (10* frac (evalf (Pi*10^(n-1))))

end:

P:= proc(n) option remember; local i, x;

if n=0 then u(0)

else unapply

(expand (u(n) +x *add (u(i) *P(n-i-1)(x), i=0..n-1)), x)

fi

end:

a:= n-> (P(n)(1) +(-1)^n*P(n)(-1))/2:

seq (a(n), n=0..30);

See A141411 for another version.

Sequence in context: A027096 A148964 A148965 this_sequence A148966 A127927 A148967

Adjacent sequences: A130617 A130618 A130619 this_sequence A130621 A130622 A130623

nonn,easy

Paul Curtz (bpcrtz(AT)free.fr), Jun 18 2007

Edited by N. J. A. Sloane, Aug 26 2009

Definition corrected, Maple program and more terms from Alois Heinz (heinz(AT)hs-heilbronn.de), Sep 06 2009