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A141904
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Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).
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3
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1, -1, 1, 1, -2, 1, -1, 23, -1, 1, 1, -44, 14, -4, 1, -1, 563, -818, 22, -5, 1, 1, -3254, 141, -1436, 19, -2, 1, -1, 88069, -13063, 21757, -457, 43, -7, 1, 1, -11384, 16774564, -11368, 7474, -680, 56, -8, 1, -1, 1593269, -1057052, 35874836, -261502, 3982, -688, 212, -3, 1, 1, -15518938, 4651811
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*sum_{i=0..n-1} u(i)*P(n-i-1,x)
and coefficients u(i)=(-1)^i/(2i+1). These u are reminiscent of the Leipniz'
Taylor expansion to calculate arctan(1) =pi/4 = A003881. Then P(n,x) = sum_{k=0..n} c(n,k)*x^k.
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REFERENCES
| P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p.44.
P. Flajolet, X. Gourdon, B. Salvy, Gazette des Mathematciens, 1993, no. 55, pp.67-78.
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EXAMPLE
| The polynomials P(n,x) are for n=0 to 5:
1 = P(0,x).
-1/3+x = P(1,x).
1/5-2/3*x+x^2 = P(2,x).
-1/7+23/45*x-x^2+x^3 = P(3,x).
1/9-44/105*x+14/15*x^2-4/3*x^3+x^4 = P(4,x).
-1/11+563/1575*x-818/945*x^2+22/15*x^3-5/3*x^4+x^5 = P(5,x).
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MAPLE
| u := proc(i) (-1)^i/(2*i+1) ; end:
P := proc(n, x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i, x), i=0..n-1) ; expand(%) ; fi; end:
A141904 := proc(n, k) p := P(n, x) ; numer(coeftayl(p, x=0, k)) ; end: seq(seq(A141904(n, k), k=0..n), n=0..13) ; # R. J. Mathar, Aug 24 2009
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CROSSREFS
| Cf. A142048 (denominators), A140749, A141412 (where u=(-1)^i/(i+1)).
Sequence in context: A156889 A172177 A156725 * A147802 A093076 A132454
Adjacent sequences: A141901 A141902 A141903 * A141905 A141906 A141907
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KEYWORD
| sign,frac,tabl
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AUTHOR
| Paul Curtz (bpcrtz(AT)free.fr), Sep 14 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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