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
KEYWORD
AUTHOR
Paul Curtz, Aug 04 2008
EXTENSIONS
Partially edited by R. J. Mathar, Aug 24 2009
STATUS
approved