A140749 Triangle c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

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

Offset: 0

Author

Paul Curtz, Jul 13 2008

Keywords

• sign
• frac
• tabl

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.

Examples

			The polynomials, for n =0,1,2, ..., are
  P(0, x) = 1;
  P(1, x) = -1/2 + x;
  P(2, x) = 1/3 - x + x^2;
  P(3, x) = -1/4 + 11/12*x - 3/2*x^2 + x^3;
  P(4, x) = 1/5 - 5/6*x + 7/4*x^2 - 2*x^3 + x^4;
  P(5, x) = -1/6 + 137/180*x - 15/8*x^2 + 17/6*x^3 - 5/2*x^4 + x^5;
and the coefficients are
   1;
  -1/2,   1;
   1/3,  -1,       1;
  -1/4,  11/12,   -3/2,   1;
   1/5,  -5/6,     7/4,  -2,     1;
  -1/6, 137/180, -15/8,  17/6,  -5/2,  1;
   1/7,  -7/10,   29/15, -7/2,  25/6, -3,   1;.
		

References

• Paul Curtz, Gazette des Mathématiciens, 1992, 52, p. 44.
• Paul Curtz, Intégration Numérique .. Note 12 du Centre de Calcul Scientifique de l'Armement, Arcueil, 1969. Now in 35170, Bruz.
• P. Flajolet, X. Gourdon, and B. Salvy, Sur une famille de polynômes issus de l'analyse numérique, Gazette des Mathématiciens, 1993, 55, pp. 67-78.

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened
• Jean-François Alcover, Plot showing roots of P(200,x) in shape of a cardioid

Crossrefs

Cf. A048594, A129891, A141412 (denominators).

Programs

Magma

[Numerator(Factorial(k+1)*StirlingFirst(n+1,k+1)/Factorial(n+1) ): k in [0..n], n in [0..12]]; // G. C. Greubel, Oct 24 2023

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 *)
Table[Numerator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0,n} ]//Flatten (* G. C. Greubel, Oct 24 2023 *)

SageMath

def A048594(n,k): return (-1)^(n-k)*numerator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
flatten([[A048594(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Formula

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

Equals Numerators of A048594(n+1,k+1)/(n+1)!. - Paul Curtz, Jul 17 2008

Extensions

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