cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A133297 a(n) = n!*Sum_{k=1..n} (-1)^(k+1)*n^(n-k-1)/(n-k)!.

Original entry on oeis.org

0, 1, 1, 5, 34, 329, 4056, 60997, 1082320, 22137201, 512801920, 13269953861, 379400765184, 11877265764025, 404067857880064, 14843708906336325, 585606019079612416, 24693567694861202273, 1108343071153648926720, 52757597474618636748421, 2654611611461360017408000
Offset: 0

Views

Author

Vladeta Jovovic, Oct 17 2007

Keywords

Links

Crossrefs

Cf. A001865 (Gamma(n, n)/exp(-n)).
Cf. A000272, A000435, A277458, A292977.

Programs

  • GAP
    a:= function(n)
    if n=0 then return 0;
    else return Factorial(n)*Sum([1..n], k-> (-1)^(k+1)*n^(n-k-1)/Factorial(n-k));
    fi;
  end;
List([0..25], n-> a(n) ); # G. C. Greubel, Aug 02 2019
  • Magma
    a:= func< n | n eq 0 select 0 else Factorial(n)*(&+[(-1)^(k+1)*n^(n-k-1)/Factorial(n-k): k in [1..n]]) >;
[a(n): n in [0..25]]; // G. C. Greubel, Aug 02 2019
  • Mathematica
    Table[n!*Sum[(-1)^(k+1)*n^(n-k-1)/(n-k)!, {k,n}], {n,0,25}] (* Stefan Steinerberger, Oct 19 2007 *)
With[{m=25}, CoefficientList[Series[Log[1-LambertW[-x]], {x,0,m}], x]*Range[0,m]!] (* G. C. Greubel, Aug 02 2019 *)
  • PARI
    my(x='x+O('x^25)); concat([0], Vec(serlaplace( log(1-lambertw(-x)) ))) \\ G. C. Greubel, Aug 02 2019
  • SageMath
    def a(n):
    if (n==0): return 0
    else: return factorial(n)*sum((-1)^(k+1)*n^(n-k-1)/factorial(n-k) for k in (1..n))
[a(n) for n in (0..25)] # G. C. Greubel, Aug 02 2019

Formula

E.g.f.: log(1-LambertW(-x)).
a(n) ~ n^(n-1)/2. - Vaclav Kotesovec, Sep 25 2013
Conjecture: a(n) = (n-1)!*( Sum_{k >= 0} (-1)^k * n^(n+k)/(n+k)! - (-1/e)^n ) for n >= 1. Cf. A000435. - Peter Bala, Jul 23 2021
From Thomas Scheuerle, Nov 17 2023: (Start)
This conjecture is true. Let "gamma" be the lower incomplete gamma function: gamma(n, x) = (n-1)! (1 - exp(-x)*Sum_{k = 0..n-1} x^k/k! ), then we can get the upper incomplete gamma function Gamma(n, x) = gamma(n, oo) - gamma(n, x). By inserting according the formula below, we will obtain the formula from Peter Bala.
a(n) = (-1)^(n+1)*Gamma(n, -n)/exp(n) = (-1)^(n+1)*A292977(n-1, n), for n > 0, where Gamma is the upper incomplete gamma function. (End)

Extensions

More terms from Stefan Steinerberger, Oct 19 2007

A089258 Transposed version of A080955: T(n,k) = A080955(k,n), n>=0, k>=-1.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 2, 2, 2, 1, 3, 5, 6, 9, 1, 4, 10, 16, 24, 44, 1, 5, 17, 38, 65, 120, 265, 1, 6, 26, 78, 168, 326, 720, 1854, 1, 7, 37, 142, 393, 872, 1957, 5040, 14833, 1, 8, 50, 236, 824, 2208, 5296, 13700, 40320, 133496, 1, 9, 65, 366, 1569, 5144, 13977, 37200, 109601, 362880, 1334961
Offset: 0

Views

Author

Philippe Deléham, Dec 12 2003

Keywords

Comments

Can be extended to columns with negative indices k<0 via T(n,k) = A292977(n,-k). - Max Alekseyev, Mar 06 2018

Examples

			n\k -1   0   1    2    3    4     5     6  ...
----------------------------------------------
0  | 1,  1,  1,   1,   1,   1,    1,    1, ...
1  | 0,  1,  2,   3,   4,   5,    6,    7, ...
2  | 1,  2,  5,  10,  17,  26,   37,   50, ...
3  | 2,  6, 16,  38,  78, 152,  236,  366, ...
4  | 9, 24, 65, 168, 393, 824, 1569, 2760, ...
...

Crossrefs

Columns: A000166, A000142, A000522, A010842, A053486, A053487, A080954.
Main diagonal gives A217701.
Cf. A080955, A008290, A292977.

Programs

  • Mathematica
    (* Assuming offset (0, 0): *)
T[n_, k_] := Exp[k - 1] Gamma[n + 1, k - 1];
Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten  (* Peter Luschny, Dec 24 2021 *)

Formula

For n > 0, k >= -1, T(n,k) is the permanent of the n X n matrix with k+1 on the diagonal and 1 elsewhere.
T(0,k) = 1.
T(n,k) = Sum_{j>=0} A008290(n,j) * (k+1)^j.
T(n,k) = n*T(n-1, k) + k^n .
T(n,k) = n! * Sum_{j=0..n} k^j/j!.
E.g.f. for k-th column: exp(k*x)/(1-x).
Assuming n >= 0, k >= 0: T(n, k) = exp(k-1)*Gamma(n+1, k-1). - Peter Luschny, Dec 24 2021

Extensions

Edited and changed offset for k to -1 by Max Alekseyev, Mar 08 2018

A300481 Rectangular array read by antidiagonals: a(m,n) = 2 * Integral_{t>=0} T_n((t-m)/2)*exp(-t)*dt, m>=0, n>=0, where T_n(x) is n-th Chebyshev polynomial of first kind.

Original entry on oeis.org

2, 2, 1, 2, 0, 0, 2, -1, -1, 3, 2, -2, 0, 2, 18, 2, -3, 3, 1, 7, 95, 2, -4, 8, -6, 2, 34, 592, 2, -5, 15, -25, 15, 13, 218, 4277, 2, -6, 24, -62, 82, -28, 80, 1574, 35010, 2, -7, 35, -123, 263, -269, 106, 579, 12879, 320589
Offset: 0

Views

Author

Max Alekseyev, Mar 06 2018

Keywords

Comments

Although negative values of m are not present here or in A300480, the two arrays are connected with the formula: a(m,n) = A300480(-m,n). Thus, they essentially represent two "halves" of the same array indexed by integers m.
a(m,n) is a polynomial in m of degree n.
For any integers m>=0, n>=0, 2 * Integral_{t=-m..m} T_n(t/2)*exp(-t)*dt = 4 * Integral_{z=-m/2..m/2} T_n(z)*exp(-2*z)*dz = a(m,n)*exp(m) - A300480(m,n)*exp(-m).

Examples

			Array starts with:
m=0: 2,  1,  0,    3,   18,     95,    592, ...
m=1: 2,  0, -1,    2,    7,     34,    218, ...
m=2: 2, -1,  0,    1,    2,     13,     80, ...
m=3: 2, -2,  3,   -6,   15,    -28,    106, ...
m=4: 2, -3,  8,  -25,   82,   -269,    920, ...
...

Crossrefs

Values for m<=0 are given in A300480.
Rows: A300482 (m=0), A300485 (m=1), A102761 (m=2), A300483 (m=-1), A300484 (m=-2).
Columns (up to signs and offset): A007395 (n=0), A000027 (n=1), A005563 (n=2).
Cf. A000179 (almost row m=2), A127672, A156995.

Programs

Formula

a(m,n) = A300480(-m,n) = Sum_{i=0..n} A127672(n,i) * i! * Sum_{j=0..i} (-m)^j/j!.
a(m,n) = Sum_{i=0..n} A127672(n,i) * A292977(i,m).
Showing 1-3 of 3 results.

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