A141904 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).

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

Offset: 0

Author

Paul Curtz, Sep 14 2008

Keywords

• sign
• frac
• tabl

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 Leibniz' Taylor expansion to calculate arctan(1) =pi/4 = A003881. Then P(n,x) = sum_{k=0..n} c(n,k)*x^k.

Examples

			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).
		

References

• P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p.44.
• P. Flajolet, X. Gourdon, B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp.67-78.

Links

• Jean-François Alcover, Roots of P(120,x) in the shape of a cardioid.

Crossrefs

Cf. A142048 (denominators), A140749, A141412 (where u=(-1)^i/(i+1)).

Programs

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

Mathematica

ClearAll[u, p]; u[n_] := (-1)^n/(2*n + 1); p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n - i - 1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[ p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Numerator (* Jean-François Alcover, Oct 02 2012 *)

Extensions

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