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A130620
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Defined in comments.
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4
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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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 exanmple we take {u(i)} to be 3,1,4,1,5,9,... (A000796).
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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 .
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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
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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);
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CROSSREFS
| See A141411 for another version.
Sequence in context: A027096 A148964 A148965 * A202246 A148966 A127927
Adjacent sequences: A130617 A130618 A130619 * A130621 A130622 A130623
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KEYWORD
| nonn,easy
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Jun 18 2007
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EXTENSIONS
| 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
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