A129841 -id:A129841 - 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

A129378 Row sums of coefficients of Bernoulli twin number polynomials.

Original entry on oeis.org

1, 1, 4, 20, 116, 744, 5160, 39360, 350784, 3749760, 42940800, 442713600, 4650877440, 109244298240, 2833294464000, -3487131648000, -2166903606067200, 51809012320665600, 6808619561103360000, -131306587205713920000, -26982365129174827008000, 595860034297401409536000

Offset: 0

Paul Curtz, Jun 08 2007

sign

The origin of the sequence are polynomials on pages 61 and 69 of the CCSA paper. The first few of the polynomials have been noted in the 1992 Gazette paper.
We construct Bernoulli twin numbers polynomials C(n,x) = Sum_{j=1..n} binomial(n-1,j-1)*B(j,x) where B(n,x) are the Bernoulli polynomials of A048998 and A048999 and where binomial(.,.) is the Pascal triangle A007318: C(0,x)=B(0,x); C(1,x)=B(1,x); C(2,x)=B(2,x)+B(1,x); C(3,x)=B(3,x)+2B(2,x)+B(1,x).
The triangle of coefficients [x^m] C(n,x) for rows n=0,1,2,.. and decreasing power m=n,...,0 along each row starts
  1;
  1, -1/2;
  1,    0, -1/3;
  1,  1/2, -1/2, -1/6;
The rightmost fraction in row n, that is, the absolute term C(n,0), is the Bernoulli twin number C(n) of A129826(n), i.e., C(n) = A129826(n)/(n+1)!.
If rows are multiplied by (n+1)!, the triangle becomes
    1;
    2,  -1;
    6,   0,  -2;
   24,  12, -12,  -4;
  120, 120, -60, -60, -4;
The sequence a(n) gives the row sums of this triangle. The sums of antidiagonals are 1, 2, 5, 24, 130, 828, 6056.... The first column of the inverse of the triangle is 1, 2, 3, 3, 0, (0 continued).

P. Curtz, Integration numerique ..., Note no. 12 CCSA (later CELAR), 1969. (See A129841, A129696.)
P. Curtz, Gazette des Mathematiciens, 1992, no. 52, p. 44.

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A000142, A007318, A048998, A048999, A129724, A129826.

f:= func< n | n le 2 select (-1)^Floor((n+1)/2)/(n+1) else (-1)^n*BernoulliNumber(Floor(n - (1-(-1)^n)/2)) >;
A129378:= func< n | n eq 0 select 1 else Factorial(n+1)*(f(n)+1) >;
[A129378(n): n in [0..30]]; // G. C. Greubel, Feb 01 2024

c[n_?EvenQ] := BernoulliB[n]; c[n_?OddQ] := -BernoulliB[n-1]; c[1] = -1/2; c[2] = -1/3; a[n_] := (n+1)!*(1+c[n]); a[0]=1; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 08 2012, after given formula *)

def f(n): return (-1)^((n+1)//2)/(n+1) if n<3 else (-1)^n*bernoulli(n-(n%2))
def A129378(n): return 1 if n==0 else factorial(n+1)*(f(n)+1)
[A129378(n) for n in range(31)] # G. C. Greubel, Feb 01 2024

a(n) = (n+1)!*(1 + C(n)) = A129826(n) + A000142(n+1), n>0.

Edited and extended by R. J. Mathar, Aug 06 2008

A129891 Sum of coefficients of polynomials defined in comments lines.

Original entry on oeis.org

1, 2, 4, 9, 20, 44, 96, 209, 455, 991, 2159, 4704, 10249, 22330, 48651, 105997, 230938, 503150, 1096225, 2388372, 5203604, 11337218, 24700671, 53815949, 117250109, 255455647, 556567394, 1212606837, 2641935832, 5756049469, 12540844137

Offset: 0

Paul Curtz, Jun 04 2007

nonn

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
whose row sums are the present sequence.
The alternating row sums are 1 0 0 1 0 0 0 -1 ...
The antidiagonal sums are 1 1 2 4 7 13 23 41 73 ...
The first column of the inverse matrix is 1 -1 1 -2 5 -11 25 -63 ...

Paul Curtz, Gazette des Mathématiciens, 1992, no. 52, p. 44.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Sums of coefficients of polynomials defined in A140530.

Cf. A129841, A129696, A130620.

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1-x+x^3)/(1-3*x+2*x^2-x^4) )); // G. C. Greubel, Oct 24 2023

a:= n-> (Matrix([1, 1, 0, 1]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [3, -2, 0, 1][i] else 0 fi)^n)[1, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 14 2009

u[n_ /; n < 3] = 1; u[n_] := 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] // Total, {n, 0, 30}] (* Jean-François Alcover, Oct 02 2012 *)

def A129891_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x+x^3)/(1-3*x+2*x^2-x^4) ).list()
A129891_list(40) # G. C. Greubel, Oct 24 2023

G.f.: (1-x+x^3)/(1-3*x+2*x^2-x^4). - Alois P. Heinz, Oct 14 2009

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

More terms from Alois P. Heinz, Oct 14 2009

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A129378 Row sums of coefficients of Bernoulli twin number polynomials.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A129891 Sum of coefficients of polynomials defined in comments lines.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

SageMath

Formula

Extensions