A141412 -id:A141412 - cp's OEIS frontend

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

1, 2, 7, 37, 261, 2301, 24343, 300455, 4238153, 67255273, 1185860331, 23000296155, 486655768525, 11155073325917, 275364320099807, 7282929854486431, 205462851526617489, 6158705454187353297, 195465061563672788947, 6548320737474275229347, 230922973019493881984021

Offset: 0

Michael Somos, Jun 23 2002

Polynomials from A140749/A141412 are linked to Stirling1 (see A048594, A129841, A140749). See also P. Flajolet, X. Gourdon, B. Salvy in, available on Internet, RR-1857.pdf (preprint of unavailable Gazette des Mathematiciens 55, 1993, pp. 67-78; for graph 2 see also X. Gourdon RR-1852.pdf, pp. 64-65). What is the corresponding graph for A152650/A152656 = simplified A009998/A119502 linked, via A152818, to a(n), then Stirling2? - Paul Curtz, Dec 16 2008
Denominators in rational approximations of Lambert W(1). See Ramanujan, Notebooks, volume 2, page 22: "2. If e^{-x} = x, shew that the convergents to x are 1/2, 4/7, 21/37, 148/261, &c." Numerators in A006153. - Michael Somos, Jan 21 2019
Call an element g in a semigroup a group element if g^j = g for some j > 1. Then a(n) is the number of group elements in the semigroup of partial transformations of an n-set. Hence a(n) = Sum_{k=0..n} A154372(n,k)*k!. - Geoffrey Critzer, Nov 27 2021

			G.f. = 1 + 2*x + 7*x^2 + 37*x^3 + 261*x^4 + 2301*x^5 + 24343*x^6 + ...

O. Ganyushkin and V Mazorchuk, Classical Finite Transformation Semigroups, Springer, 2009, page 70.
S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 2, see page 22.

Seiichi Manyama, Table of n, a(n) for n = 0..411
W. S. Gray and M. Thitsa, System Interconnections and Combinatorial Integer Sequences, in: System Theory (SSST), 2013 45th Southeastern Symposium on, Date of Conference: 11-11 March 2013.
G. Jiraskova and J. Shallit, The state complexity of star-complement-star, arXiv preprint arXiv:1203.5353 [cs.FL], 2012. - From _N. J. A. Sloane_, Sep 21 2012

Cf. A000110, A006153, A030178, A089148.
Cf. A048993, A052820.
Cf. A006153, A154372.

CoefficientList[Series[1/(Exp[-x]-x), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ 1 / (Exp[-x] - x), {x, 0, n}]]; (* Michael Somos, Jan 21 2019 *)
a[ n_] := If[ n < 0, 0, n! Sum[ (n - k + 1)^k / k!, {k, 0, n}]]; (* Michael Somos, Jan 21 2019 *)

{a(n) = if( n<0, 0, n! * polcoeff( 1 / (exp(-x + x * O(x^n)) - x), n))};

{a(n) = if( n<0, 0, n! * sum(k=0, n, (n-k+1)^k / k!))}; /* Michael Somos, Jan 21 2019 */

E.g.f.: 1 / (exp(-x) - x).
a(n) = n!*Sum_{k=0..n} (n-k+1)^k/k!. - Vladeta Jovovic, Aug 31 2003
a(n) = Sum_{k=0..n} (-1)^(n-k)*Stirling2(n, k)*A052820(k). - Vladeta Jovovic, Apr 12 2004
Recurrence: a(n+1) = 1 + Sum_{j=1..n} binomial(n, j)*a(j)*j. - Jon Perry, Apr 25 2005
E.g.f.: 1/(Q(0) - x) where Q(k) = 1 - x/(2*k+1 - x*(2*k+1)/(x - (2*k+2)/Q(k+1) )); (continued fraction ). - Sergei N. Gladkovskii, Apr 04 2013
a(n) ~ n!/((1+c)*c^(n+1)), where c = A030178 = LambertW(1) = 0.5671432904... - Vaclav Kotesovec, Jun 26 2013
O.g.f.: Sum_{k>=0} k!*x^k/(1 - (k + 1)*x)^(k+1). - Ilya Gutkovskiy, Oct 09 2018
a(n) = A006153(n+1)/(n+1). - Seiichi Manyama, Nov 05 2024

A152656 Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 24, 3, 2, 1, 1, 120, 3, 2, 1, 1, 1, 720, 15, 8, 3, 2, 1, 1, 5040, 45, 40, 3, 6, 1, 1, 1, 40320, 315, 80, 15, 24, 1, 2, 1, 1, 362880, 315, 560, 45, 24, 1, 6, 1, 1, 1, 3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1

Offset: 0

Paul Curtz, Dec 10 2008

a(n) is the last sequence of a trio with, first, A141412 and, second, A142048 (denominators).

			Contribution from _Vincenzo Librandi_, Dec 16 2012: (Start)
Triangle begins:
        1,
        1,    1,
        2,    1,    1,
        6,    1,    1,   1,
       24,    3,    2,   1,   1,
      120,    3,    2,   1,   1, 1,
      720,   15,    8,   3,   2, 1,  1,
     5040,   45,   40,   3,   6, 1,  1, 1,
    40320,  315,   80,  15,  24, 1,  2, 1, 1,
   362880,  315,  560,  45,  24, 1,  6, 1, 1, 1,
  3628800, 2835, 4480, 315, 144, 5, 24, 3, 2, 1, 1,
  ...
First column: A000142; second column: A049606. (End)

Vincenzo Librandi, Rows n = 0..100, flattened

Cf. A152650, A142048, A142412,

ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Denominator (* Jean-François Alcover, Oct 02 2012 *)

More terms from Jean-François Alcover, Oct 02 2012

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

A152650 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 4, 9, 4, 1, 1, 2, 9, 8, 5, 1, 1, 4, 27, 32, 25, 6, 1, 1, 4, 81, 32, 125, 18, 7, 1, 1, 8, 81, 128, 625, 36, 49, 8, 1, 1, 2, 243, 256, 625, 54, 343, 32, 9, 1, 1, 4, 729, 1024, 3125, 324, 2401, 256, 81, 10, 1

Offset: 0

Paul Curtz, Dec 10 2008

Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1,x) and coefficients u(i)=1/i!. These u are reminiscent of the Taylor expansion of exp(x). Then P(n,x) = Sum_{k=0..n} c(n,k)*x^k.
n!*P(n,x) are the row polynomials of A152818. - Peter Bala, Oct 09 2011
Conjecture: All roots of P(n,x) are real, hence negative. - Jean-François Alcover, Oct 10 2012

			The triangle c(n,k) and polynomials start in row n = 0 as:
1 = 1;
1, 1 = 1 + x;
1/2, 2, 1 = 1/2 + 2*x + x^2;
1/6, 2, 3, 1, = 1/6+2*x+3*x^2+x^3
1/24, 4/3, 9/2, 4, 1, = 1/24 + 4/3*x + 9/2*x^2 + 4*x^3 + x^4;
1/120, 2/3, 9/2, 8, 5, 1, = 1/120 + 2/3*x + 9/2*x^2 + 8*x^3 + 5*x^4 + x^5;
1/720, 4/15, 27/8, 32/3, 25/2, 6, 1, = 1/720 + 4/15*x + 27/8*x^2 + 32/3*x^3 + 25/2*x^4 + 6*x^5 + x^6;
1/5040, 4/45, 81/40, 32/3, 125/6, 18, 7, 1 = 1/5040 + 4/45*x + 81/40*x^2 + 32/3*x^3 + 125/6*x^4 + 18*x^5 + 7*x^6 + x^7;

Jean-François Alcover, Plot showing roots of P(20,x)

Cf. A152656 (denominators), A140749, A141412, A141904, A142048. A152818.

u := proc(i) 1/i! end:
P := proc(n,x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A152650 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end:
seq(seq(A152650(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Aug 24 2009

ClearAll[u, p]; u[n_] := 1/n!; p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n-i-1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Numerator (* Jean-François Alcover, Oct 02 2012 *)

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

A141904 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, -1, 23, -1, 1, 1, -44, 14, -4, 1, -1, 563, -818, 22, -5, 1, 1, -3254, 141, -1436, 19, -2, 1, -1, 88069, -13063, 21757, -457, 43, -7, 1, 1, -11384, 16774564, -11368, 7474, -680, 56, -8, 1, -1, 1593269, -1057052, 35874836, -261502, 3982, -688, 212, -3, 1, 1, -15518938, 4651811

Offset: 0

Paul Curtz, Sep 14 2008

Let the polynomials P be defined by P(0,x)=u(0), P(n,x)= u(n) + x*sum_{i=0..n-1} u(i)*P(n-i-1,x) and coefficients u(i)=(-1)^i/(2i+1). These u are reminiscent of the Leibniz' Taylor expansion to calculate arctan(1) =pi/4 = A003881. Then P(n,x) = sum_{k=0..n} c(n,k)*x^k.

			The polynomials P(n,x) are for n=0 to 5:
1 = P(0,x).
-1/3+x = P(1,x).
1/5-2/3*x+x^2 = P(2,x).
-1/7+23/45*x-x^2+x^3 = P(3,x).
1/9-44/105*x+14/15*x^2-4/3*x^3+x^4 = P(4,x).
-1/11+563/1575*x-818/945*x^2+22/15*x^3-5/3*x^4+x^5 = P(5,x).

P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p.44.
P. Flajolet, X. Gourdon, B. Salvy, Gazette des Mathematiciens, 1993, no. 55, pp.67-78.

Jean-François Alcover, Roots of P(120,x) in the shape of a cardioid.

Cf. A142048 (denominators), A140749, A141412 (where u=(-1)^i/(i+1)).

u := proc(i) (-1)^i/(2*i+1) ; end:
P := proc(n,x) option remember ; if n =0 then u(0); else u(n)+x*add( u(i)*procname(n-1-i,x),i=0..n-1) ; expand(%) ; fi; end:
A141904 := proc(n,k) p := P(n,x) ; numer(coeftayl(p,x=0,k)) ; end: seq(seq(A141904(n,k),k=0..n),n=0..13) ; # R. J. Mathar, Aug 24 2009

ClearAll[u, p]; u[n_] := (-1)^n/(2*n + 1); p[0][x_] := u[0]; p[n_][x_] := p[n][x] = u[n] + x*Sum[u[i]*p[n - i - 1][x] , {i, 0, n-1}] // Expand; row[n_] := CoefficientList[ p[n][x], x]; Table[row[n], {n, 0, 10}] // Flatten // Numerator (* Jean-François Alcover, Oct 02 2012 *)

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

A320637 Regular triangle read by rows: T(n,k) = Lcm_{m=k..n} d(n,k) where d(n,k) is the denominator of the unsigned Stirling1(n,k)*k!/n! for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 1, 1, 1, 12, 12, 2, 1, 1, 60, 12, 4, 1, 1, 1, 60, 180, 8, 6, 2, 1, 1, 420, 180, 120, 6, 6, 1, 1, 1, 840, 5040, 240, 240, 6, 4, 2, 1, 1, 2520, 5040, 15120, 240, 144, 4, 12, 1, 1, 1, 2520, 25200, 30240, 15120, 288, 240, 24, 3, 2, 1

Offset: 0

Michel Marcus, Oct 18 2018

			Triangle begins:
  1,
  1, 1,
  1, 2, 1,
  1, 6, 1, 1,
  1, 12, 12, 2, 1,
  1, 60, 12, 4, 1, 1,
  1, 60, 180, 8, 6, 2, 1,
  1, 420, 180, 120, 6, 6, 1, 1,
  ...

Abdelmalek Bedhouche and Bakir Farhi, On some products taken over the prime numbers, arXiv:2207.07957 [math.NT], 2022.
Bakir Farhi, On the derivatives of the integer-valued polynomials, arXiv:1810.07560 [math.NT], 2018. See Table 1 c(n,k) p. 15.

Cf. A000254, A003418, A141412.

d(n,k) = denominator(abs(stirling(n,k,1))*k!/n!);
T(n,k) = my(x = 1); for (m=k, n, x = lcm(x, d(m,k))); x;

Corrected by Michel Marcus, Jul 19 2022

cp's OEIS Frontend

A072597 Expansion of 1/(exp(-x) - x) as exponential generating function.

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A152656 Triangle read by rows: denominators of polynomials from A000142: P(0,x) = 1, P(n,x) = 1/n! + x*Sum_{i=0..n-1} P(n-i-1)/i!. Numerators are A152650.

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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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A152650 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of certain polynomials P(n,x) given below.

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A141904 Triangle of the numerators of coefficients c(n,k) = [x^k] P(n,x) of some polynomials P(n,x).

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A320637 Regular triangle read by rows: T(n,k) = Lcm_{m=k..n} d(n,k) where d(n,k) is the denominator of the unsigned Stirling1(n,k)*k!/n! for 0 <= k <= n.

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