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A141412 Table c(n,k) of the denominators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891. 7
1, 2, 1, 3, 1, 1, 4, 12, 2, 1, 5, 6, 4, 1, 1, 6, 180, 8, 6, 2, 1, 7, 10, 15, 2, 6, 1, 1, 8, 560, 240, 240, 6, 4, 2, 1, 9, 1260, 15120, 20, 144, 1, 12, 1, 1, 10, 12600, 672, 945, 32, 240, 8, 3, 2, 1, 11, 1260, 8400, 1512, 3024, 48, 240, 3, 1, 1, 1, 12, 166320, 100800, 64800, 12096, 12096 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Polynomials are characteristic polynomials of a particular John Couch Adams matrix.

General term: ((((-1)^(n-j))*C(j, n))*n!)*Integral (from 0 to i) (u*(u-1)*(u-2)* ... *(u-n))/(u-j)) du, i,j from 1 to n (see Flajolet et al.).

Denominators are 1, 2, 12, 24, 720 = A091137.

These polynomials come from the explicit case. The less interesting implicit case has the same denominators (see P. Curtz reference).

REFERENCES

P. Flajolet, X. Gourdon, B. Salvy, Gazette des Mathématiciens 55, 1993, p.67.

P. Curtz Integration .. note 12, C.C.S.A., Arcueil 1969, p. 61; ibid. pp. 62-65.

LINKS

Table of n, a(n) for n=0..71.

Bakir Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560 [math.NT], 2018.

FORMULA

Conjecture: T(n, k) = d(n+1, k+1), with d(n,k) = denominator(A000254(n, k)*k!/n!) where A000254 are the unsigned Stirling numbers of the 1st kind. See d(n,k) in Farhi link. - Michel Marcus, Oct 18 2018

EXAMPLE

Triangle begins:

  1,

  2, 1,

  3, 1, 1,

  4, 12, 2, 1,

  5, 6, 4, 1, 1,

  6, 180, 8, 6, 2, 1,

  7, 10, 15, 2, 6, 1, 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:

A141412 := proc(n, k) p := P(n, x) ; denom(coeftayl(p, x=0, k)) ; end: seq(seq(A141412(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}]; Denominator[ Flatten[ Table[ CoefficientList[p[n], x], {n, 0, 11}]]][[1 ;; 72]] (* Jean-François Alcover, Jun 17 2011 *)

CROSSREFS

Cf. A140749 (numerators).

Sequence in context: A209344 A294099 A209115 * A178623 A210765 A160183

Adjacent sequences:  A141409 A141410 A141411 * A141413 A141414 A141415

KEYWORD

nonn,frac,tabl,uned

AUTHOR

Paul Curtz, Aug 04 2008

EXTENSIONS

Partially edited by R. J. Mathar, Aug 24 2009

STATUS

approved

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Last modified October 24 05:19 EDT 2021. Contains 348217 sequences. (Running on oeis4.)