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A140749
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Table c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.
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5
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1, -1, 1, 1, -1, 1, -1, 11, -3, 1, 1, -5, 7, -2, 1, -1, 137, -15, 17, -5, 1, 1, -7, 29, -7, 25, -3, 1, -1, 363, -469, 967, -35, 23, -7, 1, 1, -761, 29531, -89, 1069, -9, 91, -4, 1, -1, 7129, -1303, 4523, -285, 3013, -105, 29, -9, 1, 1, -671, 16103, -7645, 31063, -781, 4781, -55, 12, -5, 1, -1, 83711, -190553
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| The polynomials P(n,x) are defined in A129891: P(0,x)=1 and
P(n,x) = (-1)^n/(n+1) + x* sum_{i=0..n-1) (-1)^i*P(n-1-i,x)/(i+1) = sum_{k=0..n} c(n,k)*x^k.
Numerators of A048594(n,k)/n!. [Paul Curtz (bpcrtz(AT)free.fr), Jul 17 2008]
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REFERENCES
| P. Curtz Gazette des Mathematiciens, 1992, 52, p.44.
P. Curtz Integration Numerique .. Note 12 du Centre de Calcul Scientifique de l'Armement, Arcueil, 1969. Now in 35170,Bruz.
P. Flajolet, X. Gourdon, B.Salvy id,1993, 55, pp.67-78.
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FORMULA
| (n+1)*c(n,k)=(n+1-k)*c(n-1,k)-n*c(n-1, k-1). [Edgrad Bavencoffe in 1992]
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EXAMPLE
| The coeffients and polynomials for n =0,1,2.. are
1; = 1
-1/2, 1; = -1/2+x
1/3, -1, 1; = 1/3-x+x^2
-1/4, 11/12, -3/2, 1; = -1/4+11/12*x-3/2*x^2+x^3
1/5, -5/6, 7/4, -2, 1; = 1/5-5/6*x+7/4*x^2-2*x^3+x^4
-1/6, 137/180, -15/8, 17/6, -5/2, 1; = -1/6+137/180*x-15/8*x^2+17/6*x^3-5/2*x^4+x^5
1/7, -7/10, 29/15, -7/2, 25/6, -3, 1;
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MAPLE
| P := proc(n, x) option remember ; if n =0 then 1; else (-1)^n/(n+1)+x*add( (-1)^i/(i+1)*procname(n-1-i, x), i=0..n-1) ; expand(%) ; fi; end:
A140749 := proc(n, k) p := P(n, x) ; numer(coeftayl(p, x=0, k)) ; end: seq(seq(A140749(n, k), k=0..n), n=0..13) ; # R. J. Mathar, Aug 24 2009
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MATHEMATICA
| p[0] = 1; p[n_] := p[n] = (-1)^n/(n+1) + x*Sum[(-1)^k*p[n-1-k] / (k+1), {k, 0, n-1}];
Numerator[ Flatten[ Table[ CoefficientList[p[n], x], {n, 0, 11}]]][[1 ;; 69]](* From Jean-François Alcover, Jun 17 2011 *)
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CROSSREFS
| Cf. A141412 (denominators).
Sequence in context: A160464 A038316 A139311 * A010188 A110089 A177415
Adjacent sequences: A140746 A140747 A140748 * A140750 A140751 A140752
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KEYWORD
| sign,frac,tabl
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Jul 13 2008
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EXTENSIONS
| Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Aug 24 2009
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