A002467 -id:A002467 - cp's OEIS frontend

A073333 Decimal expansion of 1/(e - 1) = Sum_{k >= 1} exp(-k).

5, 8, 1, 9, 7, 6, 7, 0, 6, 8, 6, 9, 3, 2, 6, 4, 2, 4, 3, 8, 5, 0, 0, 2, 0, 0, 5, 1, 0, 9, 0, 1, 1, 5, 5, 8, 5, 4, 6, 8, 6, 9, 3, 0, 1, 0, 7, 5, 3, 9, 6, 1, 3, 6, 2, 6, 6, 7, 8, 7, 0, 5, 9, 6, 4, 8, 0, 4, 3, 8, 1, 7, 3, 9, 1, 6, 6, 9, 7, 4, 3, 2, 8, 7, 2, 0, 4, 7, 0, 9, 4, 0, 4, 8, 7, 5, 0, 5, 7, 6, 5, 4, 6, 2, 0

Offset: 0

Robert G. Wilson v, Aug 22 2002

The value of the general continued fraction with the partial numerators (A000027) and the partial denominators (A000027). The value of the fractional limit of the numerators (A000166) and the denominators (A002467). Abs(A002467/(e-1)-A000166)->0. - Seiichi Kirikami, Oct 30 2011

			0.581976706869326424385002005109011558546869301075396136266787059648...

Calvin C. Clawson, Mathematical Mysteries: The Beauty and Magic of Numbers, Springer, 2013. See p. 225.
Wolfram Research, Mathematica, Version 4.1.0.0, Help Browser, under the function NSumExtraTerms

G. C. Greubel, Table of n, a(n) for n = 0..20000
Mohammad K. Azarian, A Limit Expression of 1/(e-1), Problem # 799, College Mathematics Journal, Vol. 36, No. 2, March 2005, p. 161. Solution appeared in Vol. 37, No. 2, March 2006, pp. 147-148.
Mohammad K. Azarian, Euler's Number Via Difference Equations, International Journal of Contemporary Mathematical Sciences, Vol. 7, 2012, No. 22, pp. 1095-1102.
H. W. Gould, A rearrangement of series based on a partition of the natural numbers, The Fibonacci Quarterly, Vol. 15, No. 1 (1977), pp. 67-72.
Don Redmond, The Evaluation of Integral_{x=0..1} floor(-ln(x)) dx, Problem #153, Advanced Problem Archive, Missouri State University.
Michel Waldschmidt, Continued fractions, Ecole de recherche CIMPA-Oujda, Théorie des Nombres et ses Applications, 18-29 mai 2015: Oujda (Maroc).
Eric Weisstein's World of Mathematics, Continued Fraction Constants.
Eric Weisstein's World of Mathematics, Generalized Continued Fraction.
Index entries for transcendental numbers.

Cf. A001113, A000027, A000166, A002467, A185393, A194807, A307178.

1/(Exp(1) - 1); // G. C. Greubel, Apr 09 2018

h:=x->sum(1/exp(n),n=1..x); evalf[110](h(1500)); evalf[110](h(4000));

RealDigits[N[Sum[Exp[-n], {n, 1, Infinity}], 120]][[1]]
RealDigits[1/(E - 1), 10, 120][[1]] (* Eric W. Weisstein, May 08 2013 *)

suminf(k=1,exp(-k)) \\ Charles R Greathouse IV, Oct 04 2011

1/(exp(1)-1) \\ Charles R Greathouse IV, Oct 04 2011

Equals 1/(exp(1)-1). - Joseph Biberstine (jrbibers(AT)indiana.edu), Nov 03 2004
Also the unique real solution to log(1+x) - log(x) = 1. Equals 1-1/(1+1/(exp(1)-2)). Continued fraction is [0:1, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, 1, 1, 10, ...]. - Gerald McGarvey, Aug 14 2004
Equals Sum_{n>=0} B_n/n!, where B_n is a Bernoulli number. - Fredrik Johansson, Oct 18 2006
1/(e-1) = 1/(1+2/(2+3/(3+4/(4+5/(5+...(continued fraction)))))). - Philippe Deléham, Mar 09 2013
Equals Integral_{x=0..oo} floor(x)*exp(-x). - Jean-François Alcover, Mar 20 2013
From Peter Bala, Oct 09 2013: (Start)
Equals (1/2)*Sum_{n >= 0} 1/sinh(2^n). (Gould, equation 22).
Define s(n) = Sum_{k = 1..n} 1/k! for n >= 1. Then 1/(e - 1) = 1 - Sum_{n >= 1} 1/( (n+1)!*s(n)*s(n+1) ) is a rapidly converging series of rationals (see A194807). Equivalently, 1/(e - 1) = 1 - 1!/(1*3) - 2!/(3*10) - 3!/(10*41) - ..., where [1, 3, 10, 41, ... ] is A002627.
We also have the alternating series 1/(e - 1) = 1!/(1*1) - 2!/(1*4) + 3!/(4*15) - 4!/(15*76) + ..., where [1, 1, 4, 15, 76, ...] is A002467. (End)
From Vaclav Kotesovec, Oct 13 2018: (Start)
Equals A185393 - 1.
Equals -LambertW(exp(1/(1 - exp(1))) / (1 - exp(1))).
Equals -1 - LambertW(-1, exp(1/(1 - exp(1))) / (1 - exp(1))). (End)
From Gleb Koloskov, Sep 03 2021: (Start)
Equals (coth(1/2)-1)/2 = (A307178-1)/2.
Equals 1/2 + 2*Integral_{x=0..oo} sin(x)/(exp(2*Pi*x)-1) dx.
Equals 1/2 + (1/Pi)*Integral_{x=0..1} sin(log(x)/(2*Pi))/(x-1) dx. (End)
Equals -lim_{n->oo} zeta(1-n, n)*n^(1 - n). - Vaclav Kotesovec and Peter Luschny, Nov 05 2021
Equals Integral_{x=0..1} floor(-log(x)) dx (see Redmond link). - Amiram Eldar, Oct 03 2023
Equals 1/2 + Sum_{k>=2} tanh(1/2^k)/2^k. - Antonio Graciá Llorente, Jan 21 2024

Entry revised by N. J. A. Sloane, Apr 07 2006

A293211 Triangle T(n,k) is the number of permutations on n elements with at least one k-cycle for 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 4, 3, 2, 15, 9, 8, 6, 76, 45, 40, 30, 24, 455, 285, 200, 180, 144, 120, 3186, 1995, 1400, 1260, 1008, 840, 720, 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040, 229384, 142695, 103040, 79380, 72576, 60480, 51840, 45360, 40320, 2293839, 1427895, 1030400, 793800, 653184, 604800, 518400, 453600, 403200, 362880

Offset: 1

Dennis P. Walsh, Oct 02 2017

T(n,k) is equivalent to n! minus the number of permutations on n elements with zero k-cycles (sequence A122974).

			T(n,k) (the first 8 rows):
:     1;
:     1,     1;
:     4,     3,     2;
:    15,     9,     8,    6;
:    76,    45,    40,   30,   24;
:   455,   285,   200,  180,  144,  120;
:  3186,  1995,  1400, 1260, 1008,  840,  720;
: 25487, 15855, 11200, 8820, 8064, 6720, 5760, 5040;
  ...
T(4,3)=8 since there are exactly 8 permutations on {1,2,3,4} with at least one 3-cycle: (1)(234), (1)(243), (2)(134), (2)(143), (3)(124), (3)(142), (4)(123), and (4)(132).

Dennis P. Walsh, The number of permutations with no k-cycles

Columns k=1-10 give: A002467, A027616, A027617, A029571, A029572, A029573, A029574, A029575, A029576, A029577.
Row sums give A132961.
T(n,n) gives A000142(n-1) for n>0.
T(2n,n) gives A052145.
Cf. A122974, A126074.

T:=(n,k)->n!*sum((-1)^(j+1)*(1/k)^j/j!,j=1..floor(n/k)); seq(seq(T(n,k),k=1..n),n=1..10);

Table[n!*Sum[(-1)^(j + 1)*(1/k)^j/j!, {j, Floor[n/k]}], {n, 10}, {k, n}] // Flatten (* Michael De Vlieger, Oct 02 2017 *)

T(n,k) = n! * Sum_{j=1..floor(n/k)} (-1)^(j+1)*(1/k)^j/j!.
T(n,k) = n! - A122974(n,k).
E.g.f. of column k: (1-exp(-x^k/k))/(1-x). - Alois P. Heinz, Oct 11 2017

A324564 Number T(n,k) of permutations p of [n] such that n-k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+n*[i=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 4, 1, 1, 0, 15, 7, 1, 1, 0, 76, 31, 11, 1, 1, 0, 455, 185, 60, 18, 1, 1, 0, 3186, 1275, 435, 113, 29, 1, 1, 0, 25487, 10095, 3473, 1001, 215, 47, 1, 1, 0, 229384, 90109, 31315, 9289, 2299, 406, 76, 1, 1, 0, 2293839, 895169, 313227, 95747, 24610, 5320, 763, 123, 1, 1, 0

Offset: 0

Alois P. Heinz, Mar 06 2019

Mirror image of A324563.

			Triangle T(n,k) begins:
      1;
      1,     0;
      1,     1,     0;
      4,     1,     1,     0;
     15,     7,     1,     1,      0;
     76,    31,    11,     1,      1,      0;
    455,   185,    60,    18,      1,      1,   0;
   3186,  1275,   435,   113,     29,      1,   1,  0;
  25487, 10095,  3473,  1001,    215,     47,   1,  1,  0;
  ...
Square array A(n,k) begins:
      1,     0,     0,     0,      0,      0, ...
      1,     1,     1,     1,      1,      1, ...
      1,     1,     1,     1,      1,      1, ...
      4,     7,    11,    18,     29,     47, ...
     15,    31,    60,   113,    215,    406, ...
     76,   185,   435,  1001,   2299,   5320, ...
    455,  1275,  3473,  9289,  24610,  65209, ...
   3186, 10095, 31315, 95747, 290203, 876865, ...
   ...

Alois P. Heinz, Rows n = 0..23, flattened
Wikipedia, Integer intervals
Wikipedia, Iverson bracket
Wikipedia, Permanent (mathematics)
Wikipedia, Permutation
Wikipedia, Symmetric group

Columns k=0-10 give: A002467 (for n>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630.
Diagonals of the triangle (rows of the array) n=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637.
Row sums give A000142.
T(2n,n) or A(n,n) gives A324638.
Cf. A002467, A324563.

b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, n!,
       LinearAlgebra[Permanent](Matrix(n, (i, j)->
      `if`(j>=i and k+jk+j, 1, 0)))))
    end:
# as triangle:
T:= (n, k)-> b(n, k)-b(n, k+1):
seq(seq(T(n, k), k=0..n), n=0..10);
# as array:
A:= (n, k)-> b(n+k, k)-b(n+k, k+1):
seq(seq(A(d-k, k), k=0..d), d=0..10);

b[n_, k_] := b[n, k] = If[k > n, 0, If[k == 0, n!, Permanent[Table[If[j >= i && k+j < n+i || i > k+j, 1, 0], {i, n}, {j, n}]]]];
(* as triangle: *)
T[n_, k_] := b[n, k] - b[n, k+1];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
(* as array: *)
A[n_, k_] := b[n+k, k] - b[n+k, k+1];
Table[A[d-k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* Jean-François Alcover, May 09 2019, after Alois P. Heinz *)

A118199 Number of partitions of n having no parts equal to the size of their Durfee squares.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 40, 53, 68, 89, 113, 146, 184, 234, 293, 369, 458, 572, 706, 874, 1073, 1320, 1611, 1970, 2393, 2909, 3518, 4255, 5122, 6167, 7394, 8862, 10585, 12637, 15038, 17886, 21213, 25141, 29723, 35112, 41383, 48737, 57278

Offset: 0

Emeric Deutsch, Apr 14 2006

nonn

a(n) = A118198(n,0).
From Gus Wiseman, May 21 2022: (Start)
Also the number of integer partitions of n > 0 that have a fixed point but whose conjugate does not, ranked by A353316. For example, the a(5) = 1 through a(10) = 10 partitions are:
  11111   222      322       422        522         622
          111111   2221      2222       3222        4222
                   1111111   3221       4221        5221
                             22211      22221       22222
                             11111111   32211       32221
                                        222111      42211
                                        111111111   222211
                                                    322111
                                                    2221111
                                                    1111111111
Partitions w/ a fixed point: A001522 (unproved), ranked by A352827 (cf. A352874).
Partitions w/o a fixed point: A064428 (unproved), ranked by A352826 (cf. A352873).
Partitions w/ a fixed point and a conjugate fixed point: A188674, reverse A325187, ranked by A353317.
Partitions w/o a fixed point or conjugate fixed point: A188674 (shifted).
(End)

			a(7) = 3 because we have [7] with size of Durfee square 1, [4,3] with size of Durfee square 2 and [3,3,1] with size of Durfee square 2.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from Alois P. Heinz)

Column k=0 of A118198.
A000041 counts partitions, strict A000009.
A000700 = self-conjugate partitions, ranked by A088902, complement A330644.
A002467 counts permutations with a fixed point, complement A000166.
A064410 counts partitions of crank 0, ranked by A342192.
A115720 and A115994 count partitions by Durfee square, rank stat A257990.
A238352 counts reversed partitions by fixed points, rank statistic A352822.
A238394 counts reversed partitions without a fixed point, ranked by A352830.
A238395 counts reversed partitions with a fixed point, ranked by A352872.
A352833 counts partitions by fixed points.
Cf. A114088, A300788, A325039, A350839, A352828, A352829, A352832.

g:=1+sum(x^(k^2+k)/(1-x^k)/product((1-x^i)^2,i=1..k-1),k=1..20): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=0..54);
# second Maple program::
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(add(b(k, d) *b(n-d*(d+1)-k, d-1),
                k=0..n-d*(d+1)), d=0..floor(sqrt(n))):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012

b[n_, i_] :=  b[n, i] = If[n == 0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := Sum[Sum[b[k, d]*b[n-d*(d+1)-k, d-1], {k, 0, n-d*(d+1)}], {d, 0, Floor[Sqrt[n]]}]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
pq[y_]:=Length[Select[Range[Length[y]],#==y[[#]]&]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],pq[#]>0&&pq[conj[#]]==0&]],{n,0,30}] (* a(0) = 0, Gus Wiseman, May 21 2022 *)

G.f.: 1+sum(x^(k^2+k)/[(1-x^k)*product((1-x^i)^2, i=1..k-1)], k=1..infinity).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (16*n*sqrt(3)). - Vaclav Kotesovec, Jun 12 2025

A127899 Transform related to the harmonic series.

Original entry on oeis.org

1, -2, 2, 0, -3, 3, 0, 0, -4, 4, 0, 0, 0, -5, 5, 0, 0, 0, 0, -6, 6, 0, 0, 0, 0, 0, -7, 7, 0, 0, 0, 0, 0, 0, -8, 8, 0, 0, 0, 0, 0, 0, 0, -9, 9, 0, 0, 0, 0, 0, 0, 0, 0, -10, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -11, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -12, 12

Offset: 1

Gary W. Adamson, Feb 04 2007

This transform is the inverse of a triangle in which each row has n terms of the harmonic series; i.e., the inverse of: 1; 1, 1/2; 1, 1/2, 1/3; ...
Eigensequence of the unsigned triangle = A002467 starting (1, 4, 15, 76, 455, ...). - Gary W. Adamson, Dec 29 2008
Table T(n,k) read by antidiagonals. T(1,1)=1, T(n,1) = n (for n>1), T(n,2) = -n, T(n,k) = 0, k > 2. - Boris Putievskiy, Jan 17 2013

			First few rows of the triangle are:
1;
-2, 2;
0, -3, 3;
0, 0, -4, 4;
0, 0, 0, -5, 5;
0, 0, 0, 0, -6, 6;
0, 0, 0, 0, 0, -7, 7;
...
From _Boris Putievskiy_, Jan 17 2013: (Start)
The start of the sequence as table:
1..-1..0..0..0..0..0...
1..-2..0..0..0..0..0...
2..-3..0..0..0..0..0...
3..-4..0..0..0..0..0...
4..-5..0..0..0..0..0...
5..-6..0..0..0..0..0...
6..-7..0..0..0..0..0...
...
The start of the sequence as triangle array read by rows:
1;
-1,1;
0,-2,2;
0,0,-3,3;
0,0,0,-4,4;
0,0,0,0,-5,5;
0,0,0,0,0,-6,6;
0,0,0,0,0,0,-7,7;
...
Row number r (r>4) contains (r-2) times '0', then '-r' and 'r'. (End)

Reinhard Zumkeller, Rows n = 1..100 of triangle, flattened
Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Cf. A002467.

a127899 n k = a127899_tabl !! (n-1) !! (k-1)
a127899_row n = a127899_tabl !! (n-1)
a127899_tabl = map reverse ([1] : xss) where
   xss = iterate (\(u : v : ws) -> u + 1 : v - 1 : ws ++ [0]) [2, -2]
-- Reinhard Zumkeller, Nov 14 2014

A127899 := proc(n,k)
    if k = n then
        n;
    elif k = n-1 then
        -n;
    else
        0;
    end if;
end proc:
seq(seq( A127899(n,k),k=1..n),n=1..13) ; # R. J. Mathar, Jul 19 2024

Table[Module[{t = Floor[(-1 + Sqrt[8 n - 7])/2], i}, i = n - t (t + 1)/2; Floor[(i + 2)/(t + 2)] (t + 1) (-1)^(i + t + 1)], {n, 78}] (* or *)
Table[If[n == 1, {n}, ConstantArray[0, n - 2]~Join~{-n, n}], {n, 12}] // Flatten (* Michael De Vlieger, Feb 11 2017 *)

from math import isqrt
def A127899(n): return -(isqrt(n<<3)+1>>1)**2+(m:=isqrt(n+1<<3)+1>>1)*((m<<1)-1)+(k:=isqrt(n+2<<3)+1>>1)*(1-k)>>1 # Chai Wah Wu, Jun 08 2025

Triangle, a(1) = 1; by rows, (n-2) zeros followed by -n, n.
From Boris Putievskiy, Jan 17 2013: (Start)
a(n) = floor((A002260(n)+2)/(A003056(n)+2))*(A003056(n)+1)*(-1)^(A002260(n)+A003056(n)+1),  n>0.
a(n) = floor((i+2)/(t+2))*(t+1)*(-1)^(i+t+1), where i=n-t*(t+1)/2, t=floor((-1+sqrt(8*n-7))/2). (End)
a(n) = floor(-1/2*A002024(n)^2 + A002024(n+1)^2-1/2*A002024(n+1) + 1/2*A002024(n+2) - 1/2*A002024(n+2)^2). - Brian Tenneson, Feb 10 2017

A232744 Numbers k for which the largest m such that m! divides k is odd.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 41, 42, 43, 45, 47, 49, 51, 53, 54, 55, 57, 59, 60, 61, 63, 65, 66, 67, 69, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 99, 101, 102, 103, 105

Offset: 1

Antti Karttunen, Dec 01 2013

nonn

Numbers k for which A055881(k) is odd.
Equally: Numbers k which have an even number of the trailing zeros in their factorial base representation A007623(k).
The sequence can be described in the following manner: Sequence includes all multiples of 1!, except that it excludes from those the multiples of 2!, except that it includes the multiples of 3! (6), except that it excludes the multiples of 4! (24), except that it includes the multiples of 5! (120), except that it excludes the multiples of 6! (720), except that it includes the multiples of 7! (5040), except that it excludes the multiples of 8! (40320), except that it includes the multiples of 9! (362880), and so on, ad infinitum.
The number of terms not exceeding m! for m>=1 is A002467(m). The asymptotic density of this sequence is 1 - 1/e (A068996). - Amiram Eldar, Feb 26 2021

Antti Karttunen, Table of n, a(n) for n = 1..9558

Complement: A232745. Cf. also A055881, A007623, A232741-A232743.
Analogous sequences for binary system: A003159 & A036554.
Cf. A002467, A068996.

seq[max_] := Select[Range[max!], EvenQ @ LengthWhile[Reverse @ IntegerDigits[#, MixedRadix[Range[max, 2, -1]]], #1 == 0 &] &]; seq[5] (* Amiram Eldar, Feb 26 2021 *)

a(1)=1, and for n>1, a(n) = a(n-1) + (2 - A000035(A055881(a(n-1)+1))).

A055596 Expansion of e.g.f. (2 - x - 2*exp(-x))/(1-x).

Original entry on oeis.org

1, 0, 2, 6, 32, 190, 1332, 10654, 95888, 958878, 10547660, 126571918, 1645434936, 23036089102, 345541336532, 5528661384510, 93987243536672, 1691770383660094, 32143637289541788, 642872745790835758, 13500327661607550920, 297007208555366120238

Offset: 1

Gary Detlefs, Jul 10 2000

nonn

It appears that a(n) = n*a(n-1) + 2(-1)^(n+1) and that the n-th term of any sequence of the form {A(0) =a, A(1)= b, A(n) = (n-1)(A(n-1)+A(n-2))} is A(n) = b*A000166(n) + a*A055596(n). A(n) can also be expressed as A(n) = n*A(n-1) + (2a-b)(-1)^(n+1). - Gary Detlefs, Jun 13 2009
For n>1, sum over all permutations beginning with an ascent, ending with a descent, and without double ascents on n elements and each contributes 2 to the power of the number of double descents. By symmetry, also the sum over all permutations with a descent, ending with an ascent, and no double ascents and the sum is (again) over 2 to the power of the number of double descents. - Richard Ehrenborg, Oct 08 2013
It also appears related to a Secret Santa problem: given n people getting names from an urn to give presents and putting them back in the urn if they get their own name, this seems to be the number of ways that it may fail for the last person, as he/she has no other name to get from the urn. - João Batista Souza de Oliveira, Jan 25 2016

			a(4)=6 since the 3 permutations 1432, 2431, 3421 all have one double descent and hence each contributes 2 to the sum. - _Richard Ehrenborg_, Oct 08 2013
For the Secret Santa, a(3)=2 since person 1 will get the names of either person 2 or 3. Suppose it was person 2. Person 2 will then get either person 1 or person 3. If he/she gets person 1, the draw will fail for person 3. The other case occurs when person 1 draws person 3, person 3 draws person 1 and the draw fails for person 2. - _João Batista Souza de Oliveira_, Jan 25 2016

Alois P. Heinz, Table of n, a(n) for n = 1..200
R. Ehrenborg and J. Jung, Descent pattern avoidance, Adv. in Appl. Math., 49 (2012) 375-390.
Lapo Cioni and Luca Ferrari, Preimages under the Queuesort algorithm, arXiv preprint arXiv:2102.07628 [math.CO], 2021; Discrete Math., 344 (2021), #112561.

Cf. A000166, A002467, A230071, A227918.

A055596:= func< n | Factorial(n)*(1 -2*(&+[(-1)^j/Factorial(j): j in [0..n]]) ) >;
[A055596(n): n in [1..30]]; // G. C. Greubel, Sep 06 2022

Rest[CoefficientList[Series[(2-x-2*E^(-x))/(1-x), {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 07 2013 *)

PARI
```
a(n)=if(n<2, n>0, n*a(n-1)-2*(-1)^n)
```

a(n)=if(n<1,0,n!*polcoeff((2-x-2*exp(-x+x*O(x^n)))/(1-x),n))

def A055596(n): return factorial(n)*( 2*bool(n==0) + 1 - 2*sum((-1)^j/factorial(j) for j in (0..n)) )
[A055596(n) for n in (1..30)] # G. C. Greubel, Sep 06 2022

E.g.f.: (2-x-2*exp(-x))/(1-x).
a(n) = (n-1)*(a(n-1) + a(n-2)) = 2*A006347(n-1), n>2.
a(n) = n! - 2*A000166(n), n>0.
a(n) ~ n! * (1-2*exp(-1)). - Vaclav Kotesovec, Oct 07 2013
For n>2, a(n) = floor(n! * (1-2*exp(-1)) + 1/2). - Peter Bala, Oct 08 2013
a(n+1) = 2*A002467(n) - n!. - Vaclav Kotesovec, Oct 08 2013

More terms from James Sellers, Jul 11 2000

A002468 The game of Mousetrap with n cards: the number of permutations of n cards having at least one hit after 2.

Original entry on oeis.org

0, 0, 1, 3, 13, 65, 397, 2819, 22831, 207605, 2094121, 23205383, 280224451, 3662810249, 51523391965, 776082247979, 12463259986087, 212573743211549, 3837628837381201, 73108996989052175, 1465703611456618891, 30847249002794047793, 679998362512214208901, 15668677914172813691699, 376683592679293811722735

Offset: 1

N. J. A. Sloane

The subsequence of primes begins: 3, 13, 397, 2819, no more through a(19). - Jonathan Vos Post, Feb 01 2011

R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Joerg Arndt, Table of n, a(n) for n = 1..102
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
J. Metzger, Email to N. J. A. Sloane, Apr 30 1991
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002467, A002468, A002469, A028306, etc.

a[n_] := (n-2)*(n-2)!-(n-4)*Subfactorial[n-3]-(n-3)*Subfactorial[n-2]; a[1]=a[2]=0; a[3]=1; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Dec 12 2014 *)

a(n) = A001563(n) - A002469(n+2). (corrected by Sean A. Irvine and Joerg Arndt, Feb 10 2014)

Added two more terms, Joerg Arndt, Feb 15 2014

A002469 The game of Mousetrap with n cards: the number of permutations of n cards in which 2 is the only hit.

Original entry on oeis.org

0, 0, 1, 5, 31, 203, 1501, 12449, 114955, 1171799, 13082617, 158860349, 2085208951, 29427878435, 444413828821, 7151855533913, 122190894996451, 2209057440250799, 42133729714051825, 845553296311189109, 17810791160738752207, 392911423093684031099

Offset: 2

N. J. A. Sloane

			G.f.: x^4 + 5*x^5 + 31*x^6 + 203*x^7 + 1501*x^8 + 12449*x^9 + 114955*x^10 + ...

R. K. Guy, Unsolved Problems Number Theory, E37.
R. K. Guy and R. J. Nowakowski, "Mousetrap," in D. Miklos, V. T. Sos and T. Szonyi, eds., Combinatorics, Paul Erdős is Eighty. Bolyai Society Math. Studies, Vol. 1, pp. 193-206, 1993.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 2..100
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
J. Metzger, Email to N. J. A. Sloane, Apr 30 1991
Daniel J. Mundfrom, A problem in permutations: the game of 'Mousetrap'. European J. Combin. 15 (1994), no. 6, 555-560.
M. Z. Spivey, Staircase rook polynomials and Cayley's game of mousetrap, Eur. J. Combinat. 30 (2) (2009) 532-539
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002468, A002467, A028306, A159610, A000255, A000166.

A002469:=n->(n-3)*floor(((n-2)!+1)/exp(1)) + (n-4)*floor(((n-3)!+1)/exp(1)): 0, seq(A002469(n), n=3..30); # Wesley Ivan Hurt, Jan 10 2017

Join[{0},Table[(n-3)Floor[((n-2)!+1)/E]+(n-4)Floor[((n-3)!+1)/E], {n,3,30}]] (* Harvey P. Dale, Feb 05 2012 *)
a[n_] := (n-3)*Subfactorial[n-2]+(n-4)*Subfactorial[n-3]; a[n_ /; n <= 3] = 0; Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Dec 12 2014 *)

default(realprecision,200);
e=exp(1);
A002469(n) = if( n<=3, 0, (n-3)*floor(((n-2)!+1)/e) + (n-4)*floor(((n-3)!+1)/e) );
/* Joerg Arndt, Apr 22 2013 */

a(n) = sum of terms in (n-2)-nd row of triangle A159610; equivalent to: a(n) = (n-2)*A000255(n-1) + A000166(n). -  Gary W. Adamson, Apr 17 2009
a(n) = (n-3)* A000166(n-2) + (n-4)* A000166(n-3). - Gary Detlefs, Apr 10 2010
a(n) = (n-3)*floor(((n-2)!+1)/e) + (n-4)*floor(((n-3)!+1)/e), for n>2. - Gary Detlefs, Apr 10 2010
G.f.: x - 1 + (1-2*x)/(x*Q(0)), where Q(k) = 1/x - (2*k+1) - (k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Apr 25 2013

More terms from Harvey P. Dale, Feb 05 2012

A248925 Triangle in which row n consists of the coefficients in Sum_{m=0..n} x^m * Product_{k=m+1..n} (1-k*x), as read by rows.

Original entry on oeis.org

1, 1, 0, 1, -2, 1, 1, -5, 7, -2, 1, -9, 27, -30, 9, 1, -14, 72, -165, 159, -44, 1, -20, 156, -597, 1149, -998, 265, 1, -27, 296, -1689, 5328, -9041, 7251, -1854, 1, -35, 512, -4057, 18840, -51665, 79579, -59862, 14833, 1, -44, 827, -8665, 55353, -221225, 544564, -776073, 553591, -133496

Offset: 0

Paul D. Hanna, Oct 16 2014

If m=n, we have Sum_{k=0..n} A008277(n, k) = A000110(n) = Sum_{j=0..n} T(n,j)*A008277(2n-j,n) where A000110(n) is the n-th Bell number. - Robert A. Russell, Apr 08 2018

			Triangle begins:
1;
1, 0;
1, -2, 1;
1, -5, 7, -2;
1, -9, 27, -30, 9;
1, -14, 72, -165, 159, -44;
1, -20, 156, -597, 1149, -998, 265;
1, -27, 296, -1689, 5328, -9041, 7251, -1854;
1, -35, 512, -4057, 18840, -51665, 79579, -59862, 14833;
1, -44, 827, -8665, 55353, -221225, 544564, -776073, 553591, -133496;
1, -54, 1267, -16935, 142003, -774755, 2756814, -6221713, 8314321, -5669406, 1334961; ...
Generating method for row n:
n=0: 1 = 1;
n=1: 1 + 0*x = (1-x) * ( 1 + x/(1-x) );
n=2: 1 - 2*x + x^2 = (1-x)*(1-2*x) * ( 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) );
n=3: 1 - 5*x + 7*x^2 - 2*x^3 = (1-x)*(1-2*x)*(1-3*x) * ( 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) );
n=4: 1 - 9*x + 27*x^2 - 30*x^3 + 9*x^4 = (1-x)*(1-2*x)*(1-3*x)*(1-4*x) * ( 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) + x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) ); ...
Compare the row g.f.s to the o.g.f. of Bell numbers (A000110):
B(x) = 1 + x/(1-x) + x^2/((1-x)*(1-2*x)) + x^3/((1-x)*(1-2*x)*(1-3*x)) + x^4/((1-x)*(1-2*x)*(1-3*x)*(1-4*x)) +...
Central terms of triangle begin:
[1, -2, 27, -597, 18840, -774755, 39320575, -2375828028, 166592007731, -13300276081039, 1191315248017730, ...].

Cf. A000166, A002467, A000110.

Table[LinearSolve[Table[StirlingS2[m+j, n], {m, 0, n}, {j, n, 0, -1}],
  Table[Sum[StirlingS2[m, j], {j, 0, n}], {m, 0, n}]], {n, 0, 20}]
  // Flatten (* Robert A. Russell, Mar 30 2018 *)
Table[PadRight[CoefficientList[Sum[x^m*Product[1-j*x, {j, m+1, n}],
  {m, 0, n}], x], n+1], {n, 0, 20}] // Flatten (* Robert A. Russell, Apr 08 2018 *)
T[n_, 0] := T[n,0] = 1;
T[n_, k_] := T[n,k] = If[kRobert A. Russell, Apr 25 2018 *)

{T(n,k)=polcoeff(sum(m=0,n, x^m*prod(j=m+1,n,1-j*x)), k)}
for(n=0,10,for(k=0,n,print1(T(n,k),", "));print(""))

Right border equals A000166, the subfactorial numbers.
Row sums equal A000166 (shift right 1 place).
Row sums of unsigned terms yields A002467(n) = n! - A000166(n).
Sum_{k=0..n} A008277(m, k) = Sum_{j=0..n} T(n, j)*A008277(m+n-j, n) where A008277(m, k) are Stirling subset numbers. - Robert A. Russell, Mar 30 2018
T(n,0) = 1.
For k>0, T(n,k) = [k==n] + [kRobert A. Russell, Apr 25 2018

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