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A140749 Triangle c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.

%I A140749 #29 Jan 31 2025 08:25:01
%S A140749 1,-1,1,1,-1,1,-1,11,-3,1,1,-5,7,-2,1,-1,137,-15,17,-5,1,1,-7,29,-7,
%T A140749 25,-3,1,-1,363,-469,967,-35,23,-7,1,1,-761,29531,-89,1069,-9,91,-4,1,
%U A140749 -1,7129,-1303,4523,-285,3013,-105,29,-9,1,1,-671,16103,-7645,31063,-781,4781,-55,12,-5,1
%N A140749 Triangle c(n,k) of the numerators of coefficients [x^k] P(n,x) of the polynomials P(n,x) of A129891.
%C A140749 The polynomials P(n,x) are defined in A129891: P(0,x)=1 and
%C A140749 P(n,x) = (-1)^n/(n+1) + x* Sum_{i=0..n-1} (-1)^i*P(n-1-i,x)/(i+1) = Sum_{k=0..n} c(n,k)*x^k.
%D A140749 Paul Curtz, Gazette des Mathématiciens, 1992, 52, p. 44.
%D A140749 Paul Curtz, Intégration Numérique .. Note 12 du Centre de Calcul Scientifique de l'Armement, Arcueil, 1969. Now in 35170, Bruz.
%D A140749 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 A140749 G. C. Greubel, <a href="/A140749/b140749.txt">Rows n = 0..50 of the triangle, flattened</a>
%H A140749 Jean-François Alcover, <a href="/A140749/a140749.pdf">Plot showing roots of P(200,x) in shape of a cardioid</a>
%F A140749 (n+1)*c(n,k) = (n+1-k)*c(n-1,k) - n*c(n-1, k-1). [Edgard Bavencoffe in 1992]
%F A140749 Equals Numerators of A048594(n+1,k+1)/(n+1)!. - _Paul Curtz_, Jul 17 2008
%e A140749 The polynomials, for n =0,1,2, ..., are
%e A140749   P(0, x) = 1;
%e A140749   P(1, x) = -1/2 + x;
%e A140749   P(2, x) = 1/3 - x + x^2;
%e A140749   P(3, x) = -1/4 + 11/12*x - 3/2*x^2 + x^3;
%e A140749   P(4, x) = 1/5 - 5/6*x + 7/4*x^2 - 2*x^3 + x^4;
%e A140749   P(5, x) = -1/6 + 137/180*x - 15/8*x^2 + 17/6*x^3 - 5/2*x^4 + x^5;
%e A140749 and the coefficients are
%e A140749    1;
%e A140749   -1/2,   1;
%e A140749    1/3,  -1,       1;
%e A140749   -1/4,  11/12,   -3/2,   1;
%e A140749    1/5,  -5/6,     7/4,  -2,     1;
%e A140749   -1/6, 137/180, -15/8,  17/6,  -5/2,  1;
%e A140749    1/7,  -7/10,   29/15, -7/2,  25/6, -3,   1;.
%p A140749 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 A140749 A140749 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end: seq(seq(A140749(n, k),k=0..n),n=0..13) ; # _R. J. Mathar_, Aug 24 2009
%t A140749 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 A140749 Numerator[ Flatten[ Table[ CoefficientList[p[n], x], {n, 0, 11}]]][[1 ;; 69]] (* _Jean-François Alcover_, Jun 17 2011 *)
%t A140749 Table[Numerator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0,n} ]//Flatten (* _G. C. Greubel_, Oct 24 2023 *)
%o A140749 (Magma) [Numerator(Factorial(k+1)*StirlingFirst(n+1,k+1)/Factorial(n+1) ): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Oct 24 2023
%o A140749 (SageMath)
%o A140749 def A048594(n,k): return (-1)^(n-k)*numerator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
%o A140749 flatten([[A048594(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Oct 24 2023
%Y A140749 Cf. A048594, A129891, A141412 (denominators).
%K A140749 sign,frac,tabl
%O A140749 0,8
%A A140749 _Paul Curtz_, Jul 13 2008
%E A140749 Edited and extended by _R. J. Mathar_, Aug 24 2009