A306234 -id:A306234 - cp's OEIS frontend

A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

0, 1, 1, 4, 15, 76, 455, 3186, 25487, 229384, 2293839, 25232230, 302786759, 3936227868, 55107190151, 826607852266, 13225725636255, 224837335816336, 4047072044694047, 76894368849186894, 1537887376983737879, 32295634916658495460, 710503968166486900119

Offset: 0

N. J. A. Sloane, Jeffrey Shallit

a(n) is the number of permutations in the symmetric group S_n that have a fixed point, i.e., they are not derangements (A000166). - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001
a(n+1)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=k! for k=0,1,...,n. - Michael Somos, Oct 07 2003
The termwise sum of this sequence and A000166 gives the factorial numbers. - D. G. Rogers, Aug 26 2006, Jan 06 2008
a(n) is the number of deco polyominoes of height n and having in the last column an odd number of cells. A deco polyomino is a directed column-convex polyomino in which the height, measured along the diagonal, is attained only in the last column. Example: a(2)=1 because the horizontal domino is the only deco polyomino of height 2 having an odd number of cells in the last column. - Emeric Deutsch, May 08 2008
Starting (1, 4, 15, 76, 455, ...) = eigensequence of triangle A127899 (unsigned). - Gary W. Adamson, Dec 29 2008
(n-1) | a(n), hence a(n) is never prime. - Jonathan Vos Post, Mar 25 2009
a(n) is the number of permutations of [n] that have at least one fixed point = number of positive terms in n-th row of the triangle in A170942, n > 0. - Reinhard Zumkeller, Mar 29 2012
Numerator of partial sum of alternating harmonic series, provided that the denominator is n!. - Richard Locke Peterson, May 11 2020
a(n) is the number of terms in the polynomial expansion of the determinant of a n X n matrix that contains at least one diagonal element. - Adam Wang, May 28 2025

			G.f. = x + x^2 + 4*x^3 + 15*x^4 + 76*x^5 + 455*x^6 + 3186*x^7 + 25487*x^8 + ...

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).

Alois P. Heinz, Table of n, a(n) for n = 0..450 (first 101 terms from T. D. Noe)
E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.
P. R. de Montmort, On the Game of Thirteen (1713), reprinted in Annotated Readings in the History of Statistics, ed. H. A. David and A. W. F. Edwards, Springer-Verlag, 2001, pp. 25-29.
Sergi Elizalde, Bijections for restricted inversion sequences and permutations with fixed points, arXiv:2006.13842 [math.CO], 2020.
R. K. Guy, Letter to N. J. A. Sloane, Feb 10 1993
R. K. Guy and R. J. Nowakowski, Mousetrap, Preprint, Feb 10 1993 [Annotated scanned copy]
R. K. Guy and S. Washburn, Correspondence, Nov. - Dec. 1991
T. Kotek, J. A. Makowsky, Recurrence Relations for Graph Polynomials on Bi-iterative Families of Graphs, arXiv preprint arXiv:1309.4020 [math.CO], 2013.
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.
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Simon Plouffe, Exact formulas for Integer Sequences.
A. Steen, Some formulas respecting the game of mousetrap, Quart. J. Pure Applied Math., 15 (1878), 230-241.
L. Takacs, The Problem of Coincidences, Archive for History of Exact Sciences, Volume 21, No. 3, Sept. 1980. pp 229-244, paragraphs 5 and 7.
Eric Weisstein's World of Mathematics, Mousetrap

Cf. A002468, A002469, A028306, A047920, A052169, A127899, A276975.
Row sums of A068106.
Column k=1 of A293211.
Column k=0 of A299789, A306234, and of A324362.

a := proc(n) -add((-1)^i*binomial(n, i)*(n-i)!, i=1..n) end;
a := n->-n!*add((-1)^k/k!, k=1..n): seq(a(n), n=0..20); # Zerinvary Lajos, May 25 2007
a := n -> simplify(GAMMA(n+1) - GAMMA(n+1, -1)*exp(-1)):
seq(a(n), n=0..20); # Peter Luschny, Feb 28 2017

Denominator[k=1; NestList[1+1/(k++ #1)&,1,12]] (* Wouter Meeussen, Mar 24 2007 *)
a[ n_] := If[ n < 0, 0, n! - Subfactorial[n]] (* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 1, 0, n! - Round[ n! / E]] (* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 0, 0, n! - (-1)^n HypergeometricPFQ[ {- n, 1}, {}, 1]](* Michael Somos, Jan 25 2014 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ (1 - Exp[ -x] ) / (1 - x), {x, 0, n}]] (* Michael Somos, Jan 25 2014 *)
RecurrenceTable[{a[n] == (n - 1) ( a[n - 1] + a[n - 2]), a[0] == 0, a[1] == 1}, a[n], {n, 20}] (* Ray Chandler, Jul 30 2015 *)

{a(n) = if( n<1, 0, n * a(n-1) - (-1)^n)} /* Michael Somos, Mar 24 2003 */

{a(n) = if( n<0, 0, n! * polcoeff( (1 - exp( -x + x * O(x^n))) / (1 - x), n))} /* Michael Somos, Mar 24 2003 */

a(n) = if(n<1,0,subst(polinterpolate(vector(n,k,(k-1)!)),x,n+1))

A002467(n) = if(n<1, 0, n*A002467(n-1)-(-1)^n); \\ Joerg Arndt, Apr 22 2013

a(n) = n! - A000166(n) = A000142(n) - A000166(n).
E.g.f.: (1 - exp(-x)) / (1 - x). - Michael Somos, Aug 11 1999
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1; a(1) = 1. - Michael Somos, Aug 11 1999
a(n) = n*a(n-1) - (-1)^n. - Michael Somos, Aug 11 1999
a(0) = 0, a(n) = floor(n!(e-1)/e + 1/2) for n > 0. - Michael Somos, Aug 11 1999
a(n) = - n! * Sum_{i=1..n} (-1)^i/i!. Limit_{n->infinity} a(n)/n! = 1 - 1/e. - Gerald McGarvey, Jun 08 2004
Inverse binomial transform of A002627. - Ross La Haye, Sep 21 2004
a(n) = (n-1)*(a(n-1) + a(n-2)), n > 1. - Gary Detlefs, Apr 11 2010
a(n) = n! - floor((n!+1)/e), n > 0. - Gary Detlefs, Apr 11 2010
For n > 0, a(n) = {(1-1/exp(1))*n!}, where {x} is the nearest integer. - Simon Plouffe, conjectured March 1993, added Feb 17 2011
0 = a(n) * (a(n+1) + a(n+2) - a(n+3)) + a(n+1) * (a(n+1) + 2*a(n+2) - a(n+3)) + a(n+2) * (a(n+2)) if n >= 0. - Michael Somos, Jan 25 2014
a(n) = Gamma(n+1) - Gamma(n+1, -1)*exp(-1). - Peter Luschny, Feb 28 2017
a(n) = Sum_{k=0..n-1} A047920(n-1,k). - Alois P. Heinz, Sep 01 2021

A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

Original entry on oeis.org

0, 1, 3, 13, 67, 411, 2921, 23633, 214551, 2160343, 23897269, 288102189, 3760013027, 52816397219, 794536751217, 12744659120521, 217140271564591, 3916221952414383, 74539067188152941, 1493136645424092773, 31400620285465593339, 691708660911435955579

Offset: 1

Emeric Deutsch, Sep 07 2010

nonn

a(n) = A180190(n,1).
a(n+2) = p(n+2) where p(x) is the unique degree-n polynomial such that p(k) = k! for k = 1, ..., n+1. - Michael Somos, Jan 05 2012
From Jon Perry, Jan 04 2013: (Start)
Number of permutations of {1,...,n-1,n+1} with at least one indexed point p(k)=k with 1<=k<=n. Note that this means p(k)=n+1 is never an indexed point as kFor n>1, a(n) is the number of permutations of [n+1] that have a fixed point and contain 12; for example the a(3)=3 such permutations of {1,2,3,4} are 1234, 1243, and 3124.
(End)
For n > 0: row sums of triangle A116853. - Reinhard Zumkeller, Aug 31 2014

			x^2 + 3*x^3 + 13*x^4 + 67*x^5 + 411*x^6 + 2921*x^7 + 23633*x^8 + ...
a(3) = 3 because we have 123, 312, and 231; the permutations 132, 213, and 321 have no successions.
a(4) = 13 since p(x) = (3*x^2 - 7*x + 6) / 2 interpolates p(1) = 1, p(2) = 2, p(3) = 6, and p(4) = 13. - _Michael Somos_, Jan 05 2012

Alois P. Heinz, Table of n, a(n) for n = 1..200
Uriel Feige, Tighter bounds for online bipartite matching, 2018.

Cf. A000142, A000166, A000255, A002467, A028417, A180190.
Cf. A116853, A276975.
Column k=1 of A306234, A306461, and of A324362(n-1).

a180191 n = if n == 1 then 0 else sum $ a116853_row (n - 1)
-- Reinhard Zumkeller, Aug 31 2014

d[0] := 1: for n to 50 do d[n] := n*d[n-1]+(-1)^n end do: seq(factorial(n)-d[n]-d[n-1], n = 1 .. 22);

f[n_] := Sum[ -(-1)^k (n - k)! Binomial[n - 1, k], {k, 1, n}]; Array[f, 20] (* Robert G. Wilson v, Oct 16 2010 *)

{a(n) = if( n<2, 0, n--; subst( polinterpolate( vector( n, k, k!)), x, n+1))} /* Michael Somos, Jan 05 2012 */

a(n) = n! - d(n) - d(n-1), where d(j) = A000166(j) are the derangement numbers.
a(n) = n! - A000255(n-1) = A002467(n) - A000166(n-1). - Jon Perry, Jan 05 2013
a(n) = (n-1)! [x^(n-1)] (1-exp(-x))/(1-x)^2. - Alois P. Heinz, Feb 23 2019

A324362 Total number of occurrences of k in the (signed) displacement sets of all permutations of [n+k] divided by k!; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 3, 4, 0, 1, 5, 13, 15, 0, 1, 7, 28, 67, 76, 0, 1, 9, 49, 179, 411, 455, 0, 1, 11, 76, 375, 1306, 2921, 3186, 0, 1, 13, 109, 679, 3181, 10757, 23633, 25487, 0, 1, 15, 148, 1115, 6576, 29843, 98932, 214551, 229384, 0, 1, 17, 193, 1707, 12151, 69299, 307833, 1006007, 2160343, 2293839

Offset: 0

Alois P. Heinz, Feb 23 2019

			Square array A(n,k) begins:
    0,    0,     0,     0,     0,      0,      0, ...
    1,    1,     1,     1,     1,      1,      1, ...
    1,    3,     5,     7,     9,     11,     13, ...
    4,   13,    28,    49,    76,    109,    148, ...
   15,   67,   179,   375,   679,   1115,   1707, ...
   76,  411,  1306,  3181,  6576,  12151,  20686, ...
  455, 2921, 10757, 29843, 69299, 142205, 266321, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened
Wikipedia, Permutation

Columns k=0-10 give: A002467, A180191(n+1), A324352, A324353, A324354, A324355, A324356, A324357, A324358, A324359, A324360.
Rows n=0-3 give: A000004, A000012, A005408, A056107(k+1).
Main diagonal gives A324361.
Cf. A306234.

A:= (n, k)-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!:
seq(seq(A(n, d-n), n=0..d), d=0..12);

m = 10;
col[k_] := col[k] = CoefficientList[(1-Exp[-x])/(1-x)^(k+1)+O[x]^(m+1), x]* Range[0, m]!;
A[n_, k_] := col[k][[n+1]];
Table[A[n, d-n], {d, 0, m}, {n, 0, d}] // Flatten (* Jean-François Alcover, May 03 2021 *)

E.g.f. of column k: (1-exp(-x))/(1-x)^(k+1).
A(n,k) = -1/k! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+k-j)!.
A(n,k) = A306234(n+k,k).

A306461 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n]; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 3, 2, 6, 10, 13, 15, 13, 10, 6, 24, 42, 56, 67, 76, 67, 56, 42, 24, 120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120, 720, 1320, 1824, 2250, 2612, 2921, 3186, 2921, 2612, 2250, 1824, 1320, 720, 5040, 9360, 13080, 16296, 19086, 21514, 23633, 25487, 23633, 21514, 19086, 16296, 13080, 9360, 5040

Offset: 1

Alois P. Heinz, Feb 17 2019

			The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement sets {p(i)-i, i=1..3}: {0}, {-1,0,1}, {-1,0,1}, {-2,1}, {-1,2}, {-2,0,2}, respectively. Numbers -2 and 2 occur twice, -1 and 1 occur thrice, and 0 occurs four times. So row n=3 is [2, 3, 4, 3, 2].
Triangle T(n,k) begins:
  :                             1                           ;
  :                        1,   1,   1                      ;
  :                   2,   3,   4,   3,   2                 ;
  :              6,  10,  13,  15,  13,  10,   6            ;
  :        24,  42,  56,  67,  76,  67,  56,  42,  24       ;
  :  120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120  ;

Alois P. Heinz, Rows n = 1..142, flattened
Wikipedia, Permutation

Columns k=0-1 give: A002467, A180191.
Row sums give A306455.
T(n+1,n) gives A000142.
T(n+2,n) gives A007680.
Cf. A000142, A061018 (left half of this triangle), A306234, A306506, A324225.

b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
      add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, {})):
seq(T(n), n=1..8);
# second Maple program:
T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n):
seq(seq(T(n, k), k=1-n..n-1), n=1..9);

T[n_, k_] := -Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 9}] // Flatten (* Jean-François Alcover, Feb 20 2021, after Alois P. Heinz *)

T(n,k) = T(n,-k).
T(n,k) = - Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.
T(n,k) = |k|! * (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).
Sum_{k=1-n..n-1} T(n,k) = A306455(n).
T(n,k) = |k|! * A306234(n,k).

A306455 Total number of covered falling diagonals in all n X n permutation matrices.

Original entry on oeis.org

0, 1, 3, 14, 73, 454, 3253, 26480, 241505, 2440538, 27075301, 327197452, 4278799105, 60205974230, 907025841317, 14567520651224, 248474458923073, 4485765986251570, 85454391074596165, 1713134893536617348, 36052727133118151201, 794697884305583064302

Offset: 0

Alois P. Heinz, Feb 16 2019

nonn

A covered diagonal in a permutation matrix contains at least one 1.
Alternatively: Total number of covered raising diagonals in all n X n permutation matrices.
Also one half of the total number of all covered diagonals in all n X n permutation matrices.
Sum over all permutations p of [n] of the cardinality of the (signed) displacement set {p(i)-i, i=1..n}.
Alternatively: Sum over all permutations p of [n] of the cardinality of the set {p(i)+i, i=1..n}.

			The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement sets {p(i)-i, i=1..3}: {0}, {-1,0,1}, {-1,0,1}, {-2,1}, {-1,2}, {-2,0,2}, representing the indices of covered falling diagonals in the permutation matrices
  [1    ]  [1    ]  [  1  ]  [  1  ]  [    1]  [    1]
  [  1  ]  [    1]  [1    ]  [    1]  [1    ]  [  1  ]
  [    1]  [  1  ]  [    1]  [1    ]  [  1  ]  [1    ] , respectively, the sum of the set cardinalities gives a(3) = 1 + 3 + 3 + 2 + 2 + 3 = 14.

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation
Wikipedia, Permutation matrix

Cf. A000142, A059841, A125182, A259834, A306234, A306461.

a:= proc(n) option remember; `if`(n<3, n*(n+1)/2,
      ((2*n^2-5*n+1)*a(n-1)-(n-1)*(n^2-4*n+2)*a(n-2)
      -(n-2)*(n-1)^2*a(n-3))/(n-2))
    end:
seq(a(n), n=0..23);

a[n_] := a[n] = If[n<3, n(n+1)/2, ((2n^2-5n+1) a[n-1] -
   (n-1)(n^2-4n+2) a[n-2] - (n-2)(n-1)^2 a[n-3])/(n-2)];
a /@ Range[0, 23] (* Jean-François Alcover, Aug 24 2021, after Alois P. Heinz *)

E.g.f.: (exp(-x)*(x+1)+x-1)/(x-1)^2.
a(n) = ((2*n^2-5*n+1)*a(n-1) - (n-1)*(n^2-4*n+2)*a(n-2) - (n-2)*(n-1)^2*a(n-3)) / (n-2) for n > 2, a(n) = n*(n+1)/2 for n < 3.
a(n) = Sum_{k=1..n} k * A125182(n,k).
a(n) = A259834(n+2) - n!.
a(n) = Sum_{k=1-n..n-1} A306461(n,k).
a(n) = Sum_{k=1-n..n-1} |k|! * A306234(n,k).
a(n) mod 2 = 1 - (n mod 2) = A059841(n) for n >= 2.

A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 4, 6, 4, 1, 1, 6, 18, 24, 18, 6, 1, 1, 8, 36, 96, 120, 96, 36, 8, 1, 1, 10, 60, 240, 600, 720, 600, 240, 60, 10, 1, 1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1, 1, 14, 126, 840, 4200, 15120, 35280, 40320, 35280, 15120, 4200, 840, 126, 14, 1

Offset: 1

Alois P. Heinz, Feb 18 2019

			Triangle T(n,k) begins:
  :                                 1                              ;
  :                           1,    2,    1                        ;
  :                     1,    4,    6,    4,    1                  ;
  :               1,    6,   18,   24,   18,    6,   1             ;
  :          1,   8,   36,   96,  120,   96,   36,   8,  1         ;
  :      1, 10,  60,  240,  600,  720,  600,  240,  60, 10,  1     ;
  :  1, 12, 90, 480, 1800, 4320, 5040, 4320, 1800, 480, 90, 12, 1  ;

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Permutation
Wikipedia, Permutation matrix

Columns k=0-6 give (offsets may differ): A000142, A001563, A001286, A005990, A061206, A062199, A062148.
Row sums give A306495(n-1).
Cf. A132159 (right part of triangle), A306234, A324225.

b:= proc(s, c) option remember; (n-> `if`(n=0, c,
      add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i)/abs(i)!, i=1-n..n-1))(b({$1..n}, 0)):
seq(T(n), n=1..8);
# second Maple program:
egf:= k-> (t-> x^t/t!*hypergeom([2, t], [t+1], x))(abs(k)+1):
T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1-n..n-1), n=1..8);
# third Maple program:
T:= (n, k)-> (t-> `if`(t

T[n_, k_] := With[{t = Abs[k]}, If[tJean-François Alcover, Mar 25 2021, after 3rd Maple program *)

T(n,k) = T(n,-k).
T(n,k) = (n-t)*(n-1)!/t! if t < n with t = |k|, T(n,k) = 0 otherwise.
T(n,k) = 1/|k|! * A324225(n,k).
E.g.f. of column k: x^t/t! * hypergeom([2, t], [t+1], x) with t = |k|+1.
Sum_{k=1-n..n-1} T(n,k) = A306495(n-1).

A324361 Total number of occurrences of n in the (signed) displacement sets of all permutations of [2n] divided by n!.

Original entry on oeis.org

0, 1, 5, 49, 679, 12151, 266321, 6906257, 206788751, 7020426511, 266464077769, 11180868467209, 513915970996583, 25678820830238759, 1385874945753239969, 80341660921985676961, 4979071555472111291551, 328496221117149603559327, 22987138271050177264124441

Offset: 0

Alois P. Heinz, Feb 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..366
Wikipedia, Permutation

Main diagonal of A324362.

Cf. A000142, A306234.

a:= proc(s) option remember; `if`(n<3, (3*n-1)*n/2,
      (8*n-12)*a(n-1)-(16*n^2-64*n+59)*a(n-2)-(4*n-10)*a(n-3))
    end:
seq(a(n), n=0..20);

A[n_, k_] := -Sum[(-1)^j*Binomial[n, j]*(n+k-j)!, {j, 1, n}]/k!;
a[n_] := A[n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Oct 28 2021, after Alois P. Heinz in A324362 *)

a(n) = n! [x^n] (1-exp(-x))/(1-x)^(n+1).
a(n) = -1/n! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (2n-j)!.
a(n) = (8*n-12)*a(n-1) - (16*n^2-64*n+59)*a(n-2) - (4*n-10)*a(n-3) for n > 2.
a(n) = A324362(n,n) = A306234(2n,n).

A324352 Total number of occurrences of 2 in the (signed) displacement sets of all permutations of [n+2] divided by 2!.

Original entry on oeis.org

0, 1, 5, 28, 179, 1306, 10757, 98932, 1006007, 11214406, 136041329, 1784556808, 25174694723, 380087428618, 6115760751869, 104481070398556, 1888837397941487, 36026457717419662, 723015306101706857, 15230427461356523056, 336009169512596054459

Offset: 0

Alois P. Heinz, Feb 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..448
Wikipedia, Permutation

Column k=2 of A324362.

Cf. A306234.

a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(2):
seq(a(n), n=0..23);

m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^3 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)

E.g.f.: (1-exp(-x))/(1-x)^3.
a(n) = -1/2! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+2-j)!.
a(n) = A306234(n+2,2).

A324353 Total number of occurrences of 3 in the (signed) displacement sets of all permutations of [n+3] divided by 3!.

Original entry on oeis.org

0, 1, 7, 49, 375, 3181, 29843, 307833, 3468671, 42432445, 560365779, 7948580377, 120557659247, 1947336998829, 33378478735475, 605158251430681, 11571369420832383, 232739737871570173, 4912330587789969971, 108564708629365952505, 2507303342099915104559

Offset: 0

Alois P. Heinz, Feb 23 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..447
Wikipedia, Permutation

Column k=3 of A324362.

Cf. A306234.

a:= n-> (k-> -add((-1)^j*binomial(n, j)*(n+k-j)!, j=1..n)/k!)(3):
seq(a(n), n=0..23);

m = 23;
CoefficientList[(1-Exp[-x])/(1-x)^4 + O[x]^(m+1), x]*Range[0, m]! (* Jean-François Alcover, May 03 2021 *)

E.g.f.: (1-exp(-x))/(1-x)^4.
a(n) = -1/3! * Sum_{j=1..n} (-1)^j * binomial(n,j) * (n+3-j)!.
a(n) = A306234(n+3,3).

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A306525 Row sums of A306234.

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A002467 The game of Mousetrap with n cards (given n letters and n envelopes, how many ways are there to fill the envelopes so that at least one letter goes into its right envelope?).

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A180191 Number of permutations of [n] having at least one succession. A succession of a permutation p is a position i such that p(i+1)-p(i) = 1.

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A324362 Total number of occurrences of k in the (signed) displacement sets of all permutations of [n+k] divided by k!; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A306461 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n]; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

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A306455 Total number of covered falling diagonals in all n X n permutation matrices.

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A324224 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices divided by |k|!; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

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