A129891 -id:A129891 - cp's OEIS frontend

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

Paul Curtz, Aug 04 2008

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).

			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;
  ...

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.

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.

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

[Denominator(Factorial(k)*StirlingFirst(n, k)/Factorial(n)): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 24 2023

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

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 *)

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

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

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

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

Original entry on oeis.org

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, 25, -3, 1, -1, 363, -469, 967, -35, 23, -7, 1, 1, -761, 29531, -89, 1069, -9, 91, -4, 1, -1, 7129, -1303, 4523, -285, 3013, -105, 29, -9, 1, 1, -671, 16103, -7645, 31063, -781, 4781, -55, 12, -5, 1

Offset: 0

Paul Curtz, Jul 13 2008

The polynomials P(n,x) are defined in A129891: P(0,x)=1 and

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.

			The polynomials, for n =0,1,2, ..., are
  P(0, x) = 1;
  P(1, x) = -1/2 + x;
  P(2, x) = 1/3 - x + x^2;
  P(3, x) = -1/4 + 11/12*x - 3/2*x^2 + x^3;
  P(4, x) = 1/5 - 5/6*x + 7/4*x^2 - 2*x^3 + x^4;
  P(5, x) = -1/6 + 137/180*x - 15/8*x^2 + 17/6*x^3 - 5/2*x^4 + x^5;
and the coefficients are
   1;
  -1/2,   1;
   1/3,  -1,       1;
  -1/4,  11/12,   -3/2,   1;
   1/5,  -5/6,     7/4,  -2,     1;
  -1/6, 137/180, -15/8,  17/6,  -5/2,  1;
   1/7,  -7/10,   29/15, -7/2,  25/6, -3,   1;.

Paul Curtz, Gazette des Mathématiciens, 1992, 52, p. 44.
Paul Curtz, Intégration Numérique .. Note 12 du Centre de Calcul Scientifique de l'Armement, Arcueil, 1969. Now in 35170, Bruz.
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.

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Jean-François Alcover, Plot showing roots of P(200,x) in shape of a cardioid

Cf. A048594, A129891, A141412 (denominators).

[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

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:
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

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}];
Numerator[ Flatten[ Table[ CoefficientList[p[n], x], {n, 0, 11}]]][[1 ;; 69]] (* Jean-François Alcover, Jun 17 2011 *)
Table[Numerator[(k+1)!*StirlingS1[n+1,k+1]/(n+1)!], {n,0,12}, {k,0,n} ]//Flatten (* G. C. Greubel, Oct 24 2023 *)

def A048594(n,k): return (-1)^(n-k)*numerator(factorial(k+1)* stirling_number1(n+1,k+1)/factorial(n+1))
flatten([[A048594(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Oct 24 2023

(n+1)*c(n,k) = (n+1-k)*c(n-1,k) - n*c(n-1, k-1). [Edgard Bavencoffe in 1992]

Equals Numerators of A048594(n+1,k+1)/(n+1)!. - Paul Curtz, Jul 17 2008

Edited and extended by R. J. Mathar, Aug 24 2009

A140530 Triangle read by rows of coefficients of polynomials defined in comments lines.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 3, 1, 3, 6, 6, 4, 1, 4, 11, 13, 10, 5, 1, 5, 18, 27, 24, 15, 6, 1, 6, 28, 51, 55, 40, 21, 7, 1

Offset: 0

Paul Curtz, Jun 04 2007

At the same time that I introduced the polynomials P(n,x) defined by P(0,x)=1 and for n>0, P(n,x)=((-1)^n)/(n+1) + x*Sum_{ i=0..n-1 } [(((-1)^i)/(i+1))*P(n-1-i,x)] (Gazette des Mathematiciens 1992), I gave the generalization P(0,x)=u(0), P(n,x) = u(n) + x*Sum_{ i=0..n-1 } u(i)*P(n-1-i,x).
For u(n), n>=0, = 1 1 1 2 3 4 5 6 7 8 ... the array of coefficients of the polynomials P(n,x) is:
1
1 1
1 2 1
2 3 3 1
3 6 6 4 1
4 11 13 10 5 1
5 18 27 24 15 6 1
6 28 51 55 40 21 7 1
...
which produces the present sequence.

P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

A129891 gives the row sums.

Edited by N. J. A. Sloane, Jul 05 2007

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

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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.

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A140530 Triangle read by rows of coefficients of polynomials defined in comments lines.

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A140512 Limit of A140485 (1, 4, 9, 13) table signed. Denominators of a sequence whose numerators are 1's.

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