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A140749 Triangle 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 (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.
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
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
Cf. A048594, A129891, A141412 (denominators).
Sequence in context: A038316 A139311 A355369 * A010188 A309389 A110089
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 March 28 05:02 EDT 2024. Contains 371235 sequences. (Running on oeis4.)