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A140749 Table c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891. 6
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; text; internal format)
OFFSET

0,8

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} binomial(n,k)*x^k.

Numerators of A048594(n,k)/n!. - Paul Curtz, Jul 17 2008

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.

LINKS

Table of n, a(n) for n=0..68.

Jean-François Alcover, Plot showing roots of P(200,x) in shape of a cardioid

FORMULA

(n+1)*c(n,k) = (n+1-k)*c(n-1,k) - n*c(n-1, k-1). [Edgard Bavencoffe in 1992]

EXAMPLE

The coefficients 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;

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

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]] (* Jean-François Alcover, Jun 17 2011 *)

CROSSREFS

Cf. A141412 (denominators).

Sequence in context: A160464 A038316 A139311 * A010188 A309389 A110089

Adjacent sequences:  A140746 A140747 A140748 * A140750 A140751 A140752

KEYWORD

sign,frac,tabl

AUTHOR

Paul Curtz, Jul 13 2008

EXTENSIONS

Edited and extended by R. J. Mathar, Aug 24 2009

STATUS

approved

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Last modified July 26 01:48 EDT 2021. Contains 346294 sequences. (Running on oeis4.)