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A130620
Defined in comments.
4
3, 9, 31, 106, 365, 1263, 4388, 15336, 53871, 190059, 673222, 2393291, 8535397, 30526712, 109449848, 393272258, 1415768769, 5105086517, 18434398665, 66647658995, 241210652738, 873773659486, 3167642169823, 11491042716338, 41708741708554, 151461799255253
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
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
See A141411 for another version.
Sequence in context: A289599 A148964 A148965 * A202246 A225340 A148966
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