%I #32 Nov 07 2023 19:46:50
%S 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,
%T 240,240,6,4,2,1,9,1260,15120,20,144,1,12,1,1,10,12600,672,945,32,240,
%U 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
%N Triangle c(n,k) of the denominators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.
%C Polynomials are characteristic polynomials of a particular John Couch Adams matrix.
%C 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.).
%C Denominators are 1, 2, 12, 24, 720 = A091137.
%C These polynomials come from the explicit case. The less interesting implicit case has the same denominators (see P. Curtz reference).
%D Paul Curtz, Intégration .. note 12, C.C.S.A., Arcueil 1969, p. 61; ibid. pp. 62-65.
%D 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.
%H G. C. Greubel, <a href="/A141412/b141412.txt">Rows n = 0..50 of the triangle, flattened</a>
%H Bakir Farhi, <a href="https://arxiv.org/abs/1810.07560">On the derivatives of the integer-valued polynomials</a>, arXiv:1810.07560 [math.NT], 2018.
%F 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
%F Equals denominators of A048594(n+1, k+1)/(n+1)!. - _G. C. Greubel_, Oct 24 2023
%e Triangle begins:
%e 1;
%e 2, 1;
%e 3, 1, 1;
%e 4, 12, 2, 1;
%e 5, 6, 4, 1, 1;
%e 6, 180, 8, 6, 2, 1;
%e 7, 10, 15, 2, 6, 1, 1;
%e ...
%p 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:
%p 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
%t 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}];
%t Denominator[Flatten[Table[CoefficientList[p[n], x], {n,0,11}]]][[1 ;; 72]] (* _Jean-François Alcover_, Jun 17 2011 *)
%t Table[Denominator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0, n}]//Flatten (* _G. C. Greubel_, Oct 24 2023 *)
%o (Magma) [Denominator(Factorial(k)*StirlingFirst(n, k)/Factorial(n)): k in [1..n], n in [1..12]]; // _G. C. Greubel_, Oct 24 2023
%o (SageMath)
%o def A141412(n,k): return denominator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
%o flatten([[A141412(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 24 2023
%Y Cf. A000254, A048594, A129891, A140749 (numerators).
%K nonn,frac,tabl
%O 0,2
%A _Paul Curtz_, Aug 04 2008
%E Partially edited by _R. J. Mathar_, Aug 24 2009