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A129891
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Sum of coefficients of polynomials defined in comments lines.
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5
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1, 2, 4, 9, 20, 44, 96, 209, 455, 991, 2159, 4704, 10249, 22330, 48651, 105997, 230938, 503150, 1096225, 2388372, 5203604, 11337218, 24700671, 53815949, 117250109, 255455647, 556567394, 1212606837, 2641935832, 5756049469, 12540844137
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| At the same time that I introduced the polynomials P(n,x) defined by P(0,x)=1 and for n>0, P(n,x)=((-1)^n)/(n+1) + x*Sum_{ i=0..n-1 } [(((-1)^i)/(i+1))*P(n-1-i,x)] (Gazette des Mathematiciens 1992), I gave the generalization P(0,x)=u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } u(i)*P(n-1-i,x).
For u(n), n>=0, = 1 1 1 2 3 4 5 6 7 8 ... the array of coefficients of the polynomials P(n,x) is:
1
1 1
1 2 1
2 3 3 1
3 6 6 4 1
4 11 13 10 5 1
5 18 27 24 15 6 1
6 28 51 55 40 21 7 1
whose row sums are the present sequence.
The alternating row sums are are 1 0 0 1 0 0 0 -1 ...
The antidiagonal sums are : 1 1 2 4 7 13 23 41 73 ...
The first column of the inverse matrix is 1 -1 1 -2 5 -11 25 -63 ...
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REFERENCES
| P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.
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FORMULA
| G.f.: -(x^3-x+1)/(x^4-2*x^2+3*x-1). [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009]
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MAPLE
| a:= n-> (Matrix([1, 1, 0, 1]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1][i] else 0 fi)^n)[1, 1]; seq (a(n), n=0..50); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009]
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CROSSREFS
| Sums of coefficients of polynomials defined in A140530.
Cf. A129841, A129696, A130620.
Sequence in context: A179744 A034007 A109975 * A130587 A129988 A035530
Adjacent sequences: A129888 A129889 A129890 * A129892 A129893 A129894
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KEYWORD
| nonn
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Jun 04 2007
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Jul 05 2007
More terms from Alois P. Heinz (heinz(AT)hs-heilbronn.de), Oct 14 2009
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