OFFSET
0,1
COMMENTS
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 example we take {u(i)} to be 3,1,4,1,5,9,... (A000796).
REFERENCES
P. Curtz, Gazette des Mathematiciens, 1992, 52, p.44.
P. Flajolet, X. Gourdon and B. Salvy, Gazette des Mathematiciens, 1993, 55, pp.67-78 .
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
a(n) ~ c * d^n, where d = 3.6412947999106071671946396356753... (same as for A141411), c = 1.38770526630795733403509218... . - Vaclav Kotesovec, Sep 12 2014
EXAMPLE
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.
MAPLE
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); # Alois P. Heinz, Sep 06 2009
MATHEMATICA
nmax = 25; digits = RealDigits[Pi, 10, nmax+1][[1]]; p[0][_] = digits[[1]]; p[n_][x_] := p[n][x] = digits[[n+1]] + x*Sum[digits[[i+1]] p[n-i-1][x], {i, 0, n-1}]; a[n_] := (p[n][1] + (-1)^n*p[n][-1])/2; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Nov 22 2012 *)
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
Paul Curtz, Jun 18 2007
EXTENSIONS
Edited by N. J. A. Sloane, Aug 26 2009
Definition corrected and more terms from Alois P. Heinz, Sep 06 2009
STATUS
approved