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

Author

Paul Curtz, Aug 04 2008

Keywords

• nonn
• frac
• tabl

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).

Examples

			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;
  ...
		

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.

Crossrefs

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

Programs

Magma

[Denominator(Factorial(k)*StirlingFirst(n, k)/Factorial(n)): k in [1..n], n in [1..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:
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 *)

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

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

Extensions

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