A140749 -id:A140749 - 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

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

Original entry on oeis.org

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

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

A231121 Denominators of coefficients of expansion of arctan(x)^3.

Original entry on oeis.org

1, 1, 15, 945, 175, 17325, 23648625, 1576575, 7309575, 1283268987, 3360942585, 1932541986375, 135664447443525, 218461268025, 242856109621125, 27604644460267875, 4479480941961650625, 1151866527932995875, 31580724596338947904875, 809762169136896100125, 4742892704944677157875

Offset: 0

Jean-François Alcover and Paul Curtz, Nov 04 2013

Cf. A002428, A002429, A071968, A140749, A141904, A142048, A081051.

a[n_] := SeriesCoefficient[ArcTan[x]^3, {x, 0, 2*n+3}] // Denominator
(* or *) a[n_] := 3*Sum[2^(i-2)*Binomial[2*(n+1), i-1]*StirlingS1[i, 3]/i!, {i, 3, 2n+3}] // Denominator; Table[a[n], {n, 0, 20}] (* from the formula given by Ruperto Corso in A002429 *)
Take[Denominator[CoefficientList[Series[ArcTan[x]^3,{x,0,50}],x] ], {4,-1,2}] (* Harvey P. Dale, Apr 07 2017 *)

Let u(n) = (-1)^n/(2*n+1) and P(n,x) = u(n) + x*Sum_{i=0..n-1} u(i)*P(n-i-1,x), with P(0,x) = u(0). Then, the terms are the denominators of the coefficients of x^2 in each polynomial.

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

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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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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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A231121 Denominators of coefficients of expansion of arctan(x)^3.

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