A130620 Defined in comments.

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

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

See A141411 for another version.

Sequence in context: A289599 A148964 A148965 * A202246 A225340 A148966

Adjacent sequences: A130617 A130618 A130619 * A130621 A130622 A130623

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