A141412 Triangle c(n,k) of the denominators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

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, 480, 360, 4, 12, 2, 1

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_{0..i} (u*(u-1)*(u-2)* ... *(u-n))/(u-j)) du, with 1 <= i,j <= 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

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

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

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

Equals denominators of A048594(n+1, k+1)/(n+1)!. - G. C. Greubel, Oct 24 2023

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

Prog

(Magma) [Denominator(Factorial(k)*StirlingFirst(n, k)/Factorial(n)): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 24 2023
(SageMath)
def A141412(n, k): return denominator(factorial(k+1)* stirling_number1(n+1, k+1)/factorial(n+1))
flatten([[A141412(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

Crossrefs

Cf. A000254, A048594, A129891, A140749 (numerators).

Sequence in context: A209115 A353430 A353391 * A178623 A210765 A362372

Adjacent sequences: A141409 A141410 A141411 * A141413 A141414 A141415

Keyword

nonn,frac,tabl

Author

Paul Curtz, Aug 04 2008

Extensions

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

Status

approved