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.
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
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
EXAMPLE
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;.
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
AUTHOR
Paul Curtz, Jul 13 2008
EXTENSIONS
Edited and extended by R. J. Mathar, Aug 24 2009
STATUS
approved