A000166 -id:A000166 - cp's OEIS frontend

A000186 Number of 3 X n Latin rectangles in which the first row is in order.

1, 0, 0, 2, 24, 552, 21280, 1073760, 70299264, 5792853248, 587159944704, 71822743499520, 10435273503677440, 1776780700509416448, 350461958856515690496, 79284041282622163140608, 20392765404792755583221760, 5917934230798104348783083520, 1924427226324694427836833857536

Offset: 0

N. J. A. Sloane

Or number of n X n matrices with exactly one 1 and one 2 in each row and column which are not in the main diagonal, other entries 0. - Vladimir Shevelev, Mar 22 2010

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 183.
Dulmage, A. L.; McMaster, G. E. A formula for counting three-line Latin rectangles. Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), pp. 279-289. Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man., 1975. MR0392611 (52 #13428). - From N. J. A. Sloane, Apr 06 2012
I. Gessel, Counting three-line Latin rectangles, Lect. Notes Math, 1234(1986), 106-111. [From Vladimir Shevelev, Mar 25 2010]
Goulden and Jackson, Combin. Enum., Wiley, 1983 p. 284.
S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127.
S. M. Kerawala, The asymptotic number of three-deep Latin rectangles, Bull. Calcutta Math. Soc., 39 (1947), 71-72.
Koichi, Yamamoto, An asymptotic series for the number of three-line Latin rectangles, J. Math. Soc. Japan 1, (1950). 226-241.
W. Moser. A generalization of Riordan's formula for 3Xn Latin rectangles, Discrete Math., 40, 311-313 [From Vladimir Shevelev, Mar 25 2010]
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210.
V. S. Shevelev, Reduced Latin rectangles and square matrices with identical sums in the rows and columns [Russian], Diskret. Mat., 4 (1992), no. 1, 91-110.
V. S. Shevelev, A generalized Riordan formula for three-rowed Latin rectangles and its applications, DAN of the Ukraine, 2 (1991), 8-12 (in Russian) [From Vladimir Shevelev, Mar 25 2010]
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).
D. S. Stones, The many formulas for the number of Latin rectangles, Electron. J. Combin 17 (2010), A1.
RJ Stones, S Lin, X Liu, G Wang, On Computing the Number of Latin Rectangles, Graphs and Combinatorics, Graphs and Combinatorics (2016) 32:1187-1202; DOI 10.1007/s00373-015-1643-1

N. J. A. Sloane, Table of n, a(n) for n = 0..207
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012. - From _N. J. A. Sloane_, Jan 02 2013
K. P. Bogart and J. Q. Longyear, Counting 3 by n Latin rectangles, Proc. Amer. Math. Soc., 54 (1976), 463-467.
S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
S. M. Kerawala, The enumeration of the Latin rectangle of depth three by means of a difference equation, Bull. Calcutta Math. Soc., 33 (1941), 119-127. [Annotated scanned copy]
S. M. Kerawala, The asymptotic number of three-deep Latin rectangles, Bull. Calcutta Math. Soc., 39 (1947), 71-72. [Annotated scanned copy]
John Riordan, Table of a(0) - a(25), circa 1968
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha, beta, gamma), arXiv preprint arXiv:1104.4051[math.CO], 2011.
D. S. Stones and I. M. Wanless, Divisors of the number of Latin rectangles, J. Combin. Theory Ser. A 117 (2010), 204-215.
Index entries for sequences related to Latin squares and rectangles

Cf. A000512, A000166, A000179, A335700, A170904.

for n from 1 to 250 do t0:=0; for j from 0 to n do for k from 0 to n-j do t0:=t0 + (2^j/j!)*k!*binomial(-3*(k+1), n-k-j); od: od: t0:=n!*t0; lprint(n,t0); od:
Maple code for A000186 based on Eq. (30) of Riordan, p. 205. Eq. (30a) on p. 206 is wrong. - N. J. A. Sloane, Jan 21 2010. Thanks to Neven Juric for correcting an error in the definition of fU, Mar 01 2010. Additional comment and modifications of code due to changes in underlying sequences from William P. Orrick, Aug 12 2020: Eq. (30) and Eq. (30a) are, in fact, related to each other by a trivial transformation and are both valid. Current code is based on Eq. (30a).
# A000166
unprotect(D);
D := proc(n) option remember; if n<=1 then 1-n else (n-1)*(D(n-1)+D(n-2));fi; end;
[seq(D(n), n=0..30)];
# A000179
U := proc(n) if n=0 then 1 else add ((-1)^k*(2*n)*binomial(2*n-k, k)*(n-k)!/(2*n-k), k=0..n); fi; end;
[seq(U(n), n=0..30)];
# A000186
K:=proc(n) local k; global D, U; add( binomial(n,k)*D(n-k)*D(k)*U(n-2*k), k=0..floor(n/2) ); end;
[seq(K(n), n=0..30)];
# another Maple program:
a:= proc(n) option remember; `if`(n<5, [1, 0, 0, 2, 24][n+1],
     ((n-1)*(n^2-2*n+2)*a(n-1) +(n-1)*(n-2)*(n^2-2*n+2)*a(n-2)
      +(n-1)*(n-2)*(n^2-2*n-2) *a(n-3)
      +2*(n-1)*(n-2)*(n-3)*(n^2-5*n+3) *a(n-4)
      -4*(n-2)*(n-3)*(n-4)*(n-1)^2 *a(n-5)) / (n-2))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Nov 02 2013

a[n_] := (t0 = 0; Do[t0 = t0 + (2^j/j!)*k!*Binomial[-3*(k+1), n-k-j], {j, 0, n}, {k, 0, n-j}]; n!*t0); Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Oct 13 2011, after Maple *)

# after Maple code based on Riordan's Eq. (30a)
d = [1,0]
for j in range(2,31):
    d.append((j - 1) * (d[-1] + d[-2]))
def u(n):
    if n == 0:
        return 1
    else:
        return sum((-1)^k * (2 * n) * binomial(2 * n - k, k) * factorial(n - k) / (2 * n - k) for k in range(0, n + 1))
def k(n):
    return sum(binomial(n, k) * d[n - k] * d[k] * u(n - 2 * k) for k in range(0, floor(n / 2) + 1))
[k(n) for n in range(0, 31)] # William P. Orrick, Aug 12 2020

a(n) = n!*Sum_{k+j<=n} (2^j/j!)*k!*binomial(-3*(k+1), n-k-j).
Note that the formula Sum_{k=0..n, k <= n/2} binomial(n, k)*D(n-k)*D(k)*U(n-2*k), where D() = A000166 and U() represents the menage numbers given by Riordan, p. 209 gives the wrong answers unless we set U(1) = -1 (or in other words we must take U() = A000179). With U(1) = 0 (see A335700) it produces A170904. See the Maple code here. - N. J. A. Sloane, Jan 21 2010, Apr 04 2010. Thanks to Vladimir Shevelev for clarifying this comment. Additional changes from William P. Orrick, Aug 12 2020
E.g.f.: exp(2*x) Sum(n>=0; n! x^n /(1+x)^(3*n+3)) from Gessel reference. - Wouter Meeussen, Nov 02 2013
a(n) ~ n!^2/exp(3). - Vaclav Kotesovec, Sep 08 2016
a(n+p)-2*a(n) is divisible by p for any prime p. - Mark van Hoeij, Jun 13 2019

Formula and more terms from Vladeta Jovovic, Mar 31 2001

Edited by N. J. A. Sloane, Jan 21 2010, Mar 04 2010, Apr 04 2010

A047920 Triangular array formed from successive differences of factorial numbers.

Original entry on oeis.org

1, 1, 0, 2, 1, 1, 6, 4, 3, 2, 24, 18, 14, 11, 9, 120, 96, 78, 64, 53, 44, 720, 600, 504, 426, 362, 309, 265, 5040, 4320, 3720, 3216, 2790, 2428, 2119, 1854, 40320, 35280, 30960, 27240, 24024, 21234, 18806, 16687, 14833, 362880, 322560

Offset: 0

N. J. A. Sloane

Number of permutations of 1,2,...,k,n+1,n+2,...,2n-k that have no agreements with 1,...,n. For example, consider 1234 and 1256, then n=4 and k=2, so T(4,2)=14. Compare A000255 for the case k=1. - Jon Perry, Jan 23 2004
From Emeric Deutsch, Apr 21 2009: (Start)
T(n-1,k-1) is the number of non-derangements of {1,2,...,n} having smallest fixed point equal to k. Example: T(3,1)=4 because we have 4213, 4231, 3214, and 3241 (the permutations of {1,2,3,4} having smallest fixed equal to 2).
Row sums give the number of non-derangement permutations of {1,2,...,n} (A002467).
Mirror image of A068106.
Closely related to A134830, where each row has an extra term (see the Charalambides reference).
(End)
T(n,k) is the number of permutations of {1..n} that don't fix the points 1..k. - Robert FERREOL, Aug 04 2016

			Triangle begins:
    1;
    1,  0;
    2,  1,  1;
    6,  4,  3,  2;
   24, 18, 14, 11,  9;
  120, 96, 78, 64, 53, 44;
  ...
The left-hand column is the factorial numbers (A000142). The other numbers in the row are calculated by subtracting the numbers in the previous row. For example, row 4 is 6, 4, 3, 2, so row 5 is 4! = 24, 24 - 6 = 18, 18 - 4 = 14, 14 - 3 = 11, 11 - 2 = 9. - _Michael B. Porter_, Aug 05 2016

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 176, Table 5.3. [From Emeric Deutsch, Apr 21 2009]

Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
E. Deutsch and S. Elizalde, The largest and the smallest fixed points of permutations, arXiv:0904.2792 [math.CO], 2009.
J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122. [Annotated scanned copy]
J. D. H. Dickson, Discussion of two double series arising from the number of terms in determinants of certain forms, Proc. London Math. Soc., 10 (1879), 120-122.
Ira M. Gessel, Symmetric inclusion-exclusion, Séminaire Lotharingien de Combinatoire, B54b (2005).
Peter Kagey, Ranking and Unranking Restricted Permutations, arXiv:2210.17021 [math.CO], 2022.
Index entries for sequences related to factorial numbers

Columns give A000142, A001563, A001564, etc. Cf. A047922.
See A068106 for another version of this triangle.
Orthogonal columns: A000166, A000255, A055790. Main diagonal A033815.
Cf. A002467, A068106, A134830. - Emeric Deutsch, Apr 21 2009
Cf. A155521.
T(n+2,n) = 2*A000153(n+1). T(n+3,n) = 6*A000261(n+2). T(n+4,n) = 24*A001909(n+3). T(n+5, n) = 120*A001910(n+4). T(n+6,n) = 720*A176732(n).
T(n+7,n) = 5040*A176733(n) - Richard R. Forberg, Dec 29 2013

a047920 n k = a047920_tabl !! n !! k
a047920_row n = a047920_tabl !! n
a047920_tabl = map fst $ iterate e ([1], 1) where
   e (row, n) = (scanl (-) (n * head row) row, n + 1)
-- Reinhard Zumkeller, Mar 05 2012

d[0] := 1: for n to 15 do d[n] := n*d[n-1]+(-1)^n end do: T := proc (n, k) if k <= n then sum(binomial(n-k, j)*d[n-j], j = 0 .. n-k) else 0 end if end proc: for n from 0 to 9 do seq(T(n, k), k = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Jul 17 2009
# second Maple program:
T:= proc(n, k) option remember;
     `if`(k=0, n!, T(n, k-1)-T(n-1, k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Sep 01 2021

t[n_, k_] = Sum[(-1)^j*Binomial[k, j]*(n-j)!, {j, 0, n}]; Flatten[Table[t[n, k], {n, 0, 9}, {k, 0, n}]][[1 ;; 47]] (* Jean-François Alcover, May 17 2011, after Philippe Deléham *)
T[n_, k_] := n! Hypergeometric1F1[-k, -n, -1];
Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten  (* Peter Luschny, Jul 28 2024 *)

row(n) = vector(n+1, k, k--; sum(j=0, n, (-1)^j * binomial(k, j)*(n-j)!)); \\ Michel Marcus, Sep 04 2021

t(n, k) = t(n, k-1) - t(n-1, k-1) = t(n, k+1) - t(n-1, k) = n*t(n-1, k) + k*t(n-2, k-1) = (n-1)*t(n-1, k-1) + (k-1)*t(n-2, k-2) = A060475(n, k)*(n-k)!. - Henry Bottomley, Mar 16 2001
T(n, k) = Sum_{j>=0} (-1)^j * binomial(k, j)*(n-j)!. - Philippe Deléham, May 29 2005
T(n,k) = Sum_{j=0..n-k} d(n-j)*binomial(n-k,j), where d(i)=A000166(i) are the derangement numbers. - Emeric Deutsch, Jul 17 2009
Sum_{k=0..n} (k+1)*T(n,k) = A155521(n+1). - Emeric Deutsch, Jul 18 2009
T(n, k) = n!*hypergeom([-k], [-n], -1). - Peter Luschny, Jul 28 2024

A055790 a(n) = na(n-1) + (n-2)a(n-2), a(0) = 0, a(1) = 2.

Original entry on oeis.org

0, 2, 4, 14, 64, 362, 2428, 18806, 165016, 1616786, 17487988, 206918942, 2657907184, 36828901754, 547499510764, 8691268384262, 146725287298888, 2624698909845026, 49592184973992676, 986871395973226286, 20630087248996393888, 451982388752415571082

Offset: 0

Henry Bottomley, Jul 13 2000

nonn

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=1 and n-1 zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, p. 201-202. - Jaap Spies, Dec 12 2003
With a(0) = 1, number of degree-(n+1) permutations p such that p(i) != i+2 for each i=1,2,...,n+1. - Vladeta Jovovic, Jan 03 2003
Equivalently number of degree-(n+1) permutations p such that p(i) != i-2 for each i=1,2,...,n+1.
With a(0) = 1, number of degree-(n+1) permutations without fixed points between 2 and n. - Olivier Gérard, Jul 29 2016
Also column 3 of Euler's difference table (third diagonal in example of A068106). - Enrique Navarrete, Oct 31 2016
For n>=2, the number of circular permutations (in cycle notation) on [n+2] that avoid substrings (j,j+3), 1<=j<=n-1.  For example, for n=2, the 4 circular permutations in S4 that avoid the substring {14} are (1234),(1243),(1324),(1342). Note that each of these circular permutations represent 4 permutations in one-line notation (see link 2017). - Enrique Navarrete, Feb 15 2017

			G.f. = 2*x + 4*x^2 + 14*x^3 + 64*x^4 + 362*x^5 + 2428*x^6 + ...
a(3) = 3*a(2)+(3-2)*a(1) = 12+2 = 14.
for n=1, the 2 permutations of [2] matches:
12, 21
for n=2, the 4 permutations of [3] without 2 as a fixed point are
132, 213, 231, 312.
for n=3, the 14 permutations of [4] without fixed point at 2 or 3 are
1324 1342 1423    2143 2314 2341 2413
3124 3142 3412 3421    4123 4312 4321

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.

Reinhard Zumkeller, Table of n, a(n) for n = 0..250
T. Mansour and M. Shattuck, Counting permutations by the number of successors within cycles, Discr. Math., 339 (2016), 1368-1376.
Enrique Navarrete, Generalized K-Shift Forbidden Substrings in Permutations, arXiv:1610.06217 [math.CO], 2016.
Enrique Navarrete, Forbidden Substrings in Circular K-Successions, arXiv:1702.02637 [math.CO], 2017.
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), pp. 197-210.

Cf. A000166 (Derangements, permutations without fixed points ).
Cf. A000255 (permutations with p(i)!=i+1, same type of recurrence).
Cf. A000153, A000261, A001909, A001910, A090010, A090012-A090016.
Apart from first term, appears in triangles A047920 or A068106 of differences of factorials, i.e. as third term of A000142, A001563, A001564, A001565 etc.

a055790 n = a055790_list !! n
a055790_list = 0 : 2 : zipWith (+)
   (zipWith (*) [0..] a055790_list) (zipWith (*) [2..] $ tail a055790_list)
-- Reinhard Zumkeller, Mar 05 2012

f := proc(n) option remember; if n <= 1 then 2*n else n*f(n-1)+(n-2)*f(n-2); fi; end;

a[0] = 0; a[1] = 2; a[n_] := a[n] = a[n] = n*a[n - 1] + (n - 2)*a[n - 2];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Nov 14 2017 *)
RecurrenceTable[{a[0]==0,a[1]==2,a[n]==n*a[n-1]+(n-2)a[n-2]},a,{n,30}] (* Harvey P. Dale, May 07 2018 *)

a(n) = if(n==0, 0, round((n+3+1/n)*n!/exp(1))) \\ Felix Fröhlich, Jul 29 2016

For n > 0, a(n) = round[(n+3+1/n)*n!/e] = 2*A000153(n) = A000255(n-1)+A000255(n) = A000166(n-1)+2*A000166(n)+A000166(n+1).
G.f.: Q(0)*(1+x)/x - 1/x - 2, where Q(k)= 1 + (2*k + 1)*x/( (1+x) - 2*x*(1+x)*(k+1)/(2*x*(k+1) + (1+x)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 08 2013
G.f.: (1+x)^2/x/Q(0) - 1/x - 2, where Q(k)= 1 - 2*k*x - x^2*(k + 1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 08 2013
G.f.: 2*(1+x)/G(0) - 1-x, where G(k)= 1 + 1/(1 - x*(2*k+2)/(x*(2*k+1) - 1 + x*(2*k+2)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 31 2013
G.f.: W(0) -1, where W(k) = 1 - x*(k+2)/( x*(k+1) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013
a(n) ~ sqrt(Pi/2)*n^n*sqrt(n)*(12*n + 37)/(6*exp(n+1)). - Ilya Gutkovskiy, Jul 29 2016
0 = a(n)*(+a(n+1) + 3*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)) for n>=0. - Michael Somos, Nov 01 2016

Comments corrected, new interpretation and examples by Olivier Gérard, Jul 29 2016

A000261 a(n) = na(n-1) + (n-3)a(n-2), with a(1) = 0, a(2) = 1.

Original entry on oeis.org

0, 1, 3, 13, 71, 465, 3539, 30637, 296967, 3184129, 37401155, 477471021, 6581134823, 97388068753, 1539794649171, 25902759280525, 461904032857319, 8702813980639617, 172743930157869827, 3602826440828270029, 78768746000235327495, 1801366114380914335441

Offset: 1

N. J. A. Sloane

nonn

With offset 1, permanent of (0,1)-matrix of size n X (n+d) with d=3 and n zeros not on a line. This is a special case of Theorem 2.3 of Seok-Zun Song et al. Extremes of permanents of (0,1)-matrices, pp. 201-202. - Jaap Spies, Dec 12 2003
a(n+2)=:b(n), n>=1, enumerates the ways to distribute n beads, labeled differently from 1 to n, over a set of (unordered) necklaces, excluding necklaces with exactly one bead, and three indistinguishable, ordered, fixed cords, each allowed to have any number of beads. Beadless necklaces as well as a beadless cords contribute each a factor 1 in the counting, e.g., b(0):= 1*1 =1. See A000255 for the description of a fixed cord with beads.
This produces for b(n) the exponential (aka binomial) convolution of the subfactorial sequence {A000166(n)} and the sequence {A001710(n+2)}. See the necklaces and cords problem comment in A000153. Therefore also the recurrence b(n) = (n+2)*b(n-1) + (n-1)*b(n-2) with b(-1)=0 and b(0)=1 holds. This comment derives from a family of recurrences found by Malin Sjodahl for a combinatorial problem for certain quark and gluon diagrams (Feb 27 2010). - Wolfdieter Lang, Jun 02 2010

			Necklaces and 3 cords problem. For n=4 one considers the following weak 2 part compositions of 4: (4,0), (3,1), (2,2), and (0,4), where (1,3) does not appear because there are no necklaces with 1 bead. These compositions contribute respectively sf(4)*1,binomial(4,3)*sf(3)*c3(1), (binomial(4,2)*sf(2))*c3(2), and 1*c3(4) with the subfactorials sf(n):=A000166(n) (see the necklace comment there) and the c3(n):=A001710(n+2) = (n+2)!/2! numbers for the pure 3 cord problem (see the remark on the e.g.f. for the k cords problem in A000153; here for k=3: 1/(1-x)^3). This adds up as 9 + 4*2*3 + (6*1)*12 + 360 = 465 = b(4) = A000261(6). - _Wolfdieter Lang_, Jun 02 2010
G.f. = x^2 + 3*x^3 + 13*x^4 + 71*x^5 + 465*x^6 + 3539*x^7 + 30637*x^8 + ...

Brualdi, Richard A. and Ryser, Herbert J., Combinatorial Matrix Theory, Cambridge NY (1991), Chapter 7.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 188.
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).

T. D. Noe, Table of n, a(n) for n=1..102
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - From _N. J. A. Sloane_, Feb 06 2013
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Special issue on the Combinatorial Matrix Theory Conference (Pohang, 2002). Linear Algebra Appl. 373 (2003), pp. 197-210.

Cf. A000255, A000153, A001909, A001910, A090010, A055790, A090012-A090016.
Cf. A086764(n+1,3), n>=1.
Cf. A000153 (necklaces and two cords). - Wolfdieter Lang, Jun 02 2010

a:= proc(n) a(n):= `if`(n<3, n-1, n*a(n-1) +(n-3)*a(n-2)) end:
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 03 2012
a := n -> `if`(n=1,0,hypergeom([4,-n+2],[],1))*(-1)^(n); seq(round(evalf(a(n), 100)), n=1..22); # Peter Luschny, Sep 20 2014

nn=20;Prepend[Range[0,nn]!CoefficientList[Series[Exp[-x]/ (1-x)^4, {x,0,nn}],x],0]  (* Geoffrey Critzer, Nov 03 2012 *)
a[ n_] := SeriesCoefficient[ x^2 HypergeometricPFQ[ {1, 4}, {}, x / (1 + x)] / (1 + x), {x, 0, n}]; (* Michael Somos, May 04 2014 *)
a[ n_] := If[ n < 2, 0, With[{m = n - 1}, Round[ Gamma[m] (m^3 + 6 m^2 + 8 m + 1) Exp[-1]/6]]]; (* Michael Somos, May 04 2014 *)

E.g.f.: exp(-x)/(1-x)^4 for offset -1.
For offset -1: (1/6)*Sum_{k=0..n} (-1)^k*(n-k+1)*(n-k+2)*(n-k+3)*n!/k! = (1/6)*(A000166(n)+3*A000166(n+1)+3*A000166(n+2)+A000166(n+3)). - Vladeta Jovovic, Jan 07 2003
a(n+1) = round( GAMMA(n)*(n^3+6*n^2+8*n+1)*exp(-1)/6 ) for n>0. - Mark van Hoeij, Nov 11 2009
G.f.: x^2*hypergeom([1,4],[],x/(x+1))/(x+1). - Mark van Hoeij, Nov 07 2011
E.g.f. for offset -1: 1/(exp(x)*(1-x)^4) = 1/E(0) where E(k) = 1 - 4*x/(1 + 3*x/(2 - 3*x + 4*x/(3 - 2*x + 3*x/(4 - x - 4/(1 + x^3*(k+1)/E(k+1)))))); (continued fraction, 3rd kind, 6-step). - Sergei N. Gladkovskii, Sep 21 2012
a(n) = hypergeometric([4,-n+2],[],1)*(-1)^n for n>=2. - Peter Luschny, Sep 20 2014

More terms from Vladeta Jovovic, Jan 07 2003

A000387 Rencontres numbers: number of permutations of [n] with exactly two fixed points.

Original entry on oeis.org

0, 0, 1, 0, 6, 20, 135, 924, 7420, 66744, 667485, 7342280, 88107426, 1145396460, 16035550531, 240533257860, 3848532125880, 65425046139824, 1177650830516985, 22375365779822544, 447507315596451070, 9397653627525472260, 206748379805560389951

Offset: 0

N. J. A. Sloane

Also: odd permutations of length n with no fixed points. - Martin Wohlgemuth (mail(AT)matroid.com), May 31 2003

Also number of cycles of length 2 in all derangements of [n]. Example: a(4)=6 because in the derangements of [4], namely (1432), (1342), (13)(24), (1423), (12)(34), (1243), (1234), (1324), and (14)(23), we have altogether 6 cycles of length 2. - Emeric Deutsch, Mar 31 2009

			a(4)=6 because we have 1243, 1432, 1324, 4231, 3214, and 2134. - _Emeric Deutsch_, Mar 31 2009

A. Kaufmann, Introduction à la combinatorique en vue des applications, Dunod, Paris, 1968 (see p. 92).
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.
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).

Chai Wah Wu, Table of n, a(n) for n = 0..200 (first 100 terms from T. D. Noe)
Bashir Ali and A. Umar, Some combinatorial properties of the alternating group, Southeast Asian Bulletin Math. 32 (2008), 823-830.
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
G. Gordon and E. McMahon, Moving faces to other places: facet derangements, Amer. Math. Monthly, 117 (2010), 865-88.
Piotr Miska, Arithmetic Properties of the Sequence of Derangements and its Generalizations, arXiv:1508.01987 [math.NT], 2015. (see Chapter 5 p. 44)
J. M. Thomas, The number of even and odd absolute permutations of n letters, Bull. Amer. Math. Soc. 31 (1925), 303-304.
M. Wohlgemuth, Derangements revisited
Index entries for sequences related to permutations with fixed points

Column k=2 of A008290.
Cf. A003221.
A diagonal of A008291.
Cf. A170942.

A000387:= n-> -add((n-1)!*add((-1)^k/(k-1)!, j=0..n-1), k=1..n-1)/2: seq(A000387(n), n=0..25); # Zerinvary Lajos, May 18 2007
A000387 := n -> (-1)^n*(hypergeom([-n,1],[],1)+n-1)/2:
seq(simplify(A000387(n)), n=0..22); # Peter Luschny, May 09 2017

Table[Subfactorial[n - 2]*Binomial[n, 2], {n, 0, 22}] (* Zerinvary Lajos, Jul 10 2009 *)

my(x='x+O('x^33)); concat([0,0], Vec( serlaplace(exp(-x)/(1-x)*(x^2/2!)) ) ) \\ Joerg Arndt, Feb 19 2014

a(n) = ( n!*sum(r=2, n, (-1)^r/r!) - (-1)^(n-1)*(n-1))/2; \\ Michel Marcus, Apr 22 2016

A145221_list, m, x = [], 1, 0
for n in range(201):
    x, m = x*n + m*(n*(n-1)//2), -m
    A145221_list.append(x) # Chai Wah Wu, Sep 23 2014

a(n) = Sum_{j=2..n-2} (-1)^j*n!/(2!*j!) = A008290(n,2).
a(n) = (n!/2) * Sum_{i=0..n-2} ((-1)^i)/i!.
a(n) = A000166(n) - A003221(n).
a(n) = A000166(n-2)*binomial(n, 2). - David Wasserman, Aug 13 2004
E.g.f.: z^2*exp(-z)/(2*(1-z)). - Emeric Deutsch, Jul 22 2009
a(n) ~ n!*exp(-1)/2. - Steven Finch, Mar 11 2022
a(n) = n*a(n-1) + (-1^n)*n*(n-1)/2, a(0) = 0. - Chai Wah Wu, Sep 23 2014
a(n) = A003221(n) + (-1)^n*(n-1) (see Miska). - Michel Marcus, Aug 11 2015
O.g.f.: (1/2)*Sum_{k>=2} k!*x^k/(1 + x)^(k+1). - Ilya Gutkovskiy, Apr 13 2017
D-finite with recurrence +(-n+2)*a(n) +n*(n-3)*a(n-1) +n*(n-1)*a(n-2)=0. - R. J. Mathar, Jul 06 2023

Prepended a(0)=a(1)=0, Joerg Arndt, Apr 22 2016

A008291 Triangle of rencontres numbers.

Original entry on oeis.org

1, 2, 3, 9, 8, 6, 44, 45, 20, 10, 265, 264, 135, 40, 15, 1854, 1855, 924, 315, 70, 21, 14833, 14832, 7420, 2464, 630, 112, 28, 133496, 133497, 66744, 22260, 5544, 1134, 168, 36, 1334961, 1334960, 667485, 222480, 55650, 11088, 1890, 240, 45, 14684570

Offset: 2

N. J. A. Sloane

T(n,k) = number of permutations of n elements with k fixed points.

T(n,n-1)=0 and T(n,n)=1 are omitted from the array. - Geoffrey Critzer, Nov 28 2011.

			Triangle begins:
       1
       2      3
       9      8     6
      44     45    20    10
     265    264   135    40   15
    1854   1855   924   315   70   21
   14833  14832  7420  2464  630  112  28
  133496 133497 66744 22260 5544 1134 168 36
  ...

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 194.
Kaufmann, Arnold. "Introduction a la combinatorique en vue des applications." Dunod, Paris, 1968. See p. 92.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 65.

T. D. Noe, Rows n=2..50, flattened
FindStat - Combinatorial Statistic Finder, The number of fixed points of a permutation
I. Kaplansky, Symbolic solution of certain problems in permutations, Bull. Amer. Math. Soc., 50 (1944), 906-914.

Cf. A008290, A170942.
Diagonals give A000217, A007290, A060008, A060836, A000166, A000240, A000387, A000449, A000475.
Row sums give A033312.
Cf. A320582.

T:= proc(n, k) T(n, k):= `if`(k=0, `if`(n<2, 1-n, (n-1)*
      (T(n-1, 0)+T(n-2, 0))), binomial(n, k)*T(n-k, 0))
    end:
seq(seq(T(n, k), k=0..n-2), n=2..12);  # Alois P. Heinz, Mar 17 2013

Prepend[Flatten[f[list_]:=Select[list,#>1&];Map[f,Drop[Transpose[Table[d = Exp[-x]/(1 - x);Range[0, 10]! CoefficientList[Series[d x^k/k!, {x, 0, 10}],x], {k, 0, 8}]], 3]]], 1] (* Geoffrey Critzer, Nov 28 2011 *)

T(n, k)= if(k<0 || k>n, 0, n!/k!*sum(i=0, n-k, (-1)^i/i!))

T(n,k) = binomial(n,k)*A000166(n-k) = A008290(n,k).
E.g.f. for column k: (x^k/k!)(exp(-x)/(1-x)). - Geoffrey Critzer, Nov 28 2011
Row generating polynomials appear to be given by -1 + sum {k = 0..n} (-1)^(n+k)*C(n,k)*(1+k*x)^(n-k)*(2+(k-1)*x)^k. - Peter Bala, Dec 29 2011

Comments and more terms from Michael Somos, Apr 26 2000

A053556 Denominator of Sum_{k=0..n} (-1)^k/k!.

Original entry on oeis.org

1, 1, 2, 3, 8, 30, 144, 280, 5760, 45360, 44800, 3991680, 43545600, 172972800, 6706022400, 93405312000, 42268262400, 22230464256000, 376610217984000, 250298560512000, 11640679464960000, 196503623737344000, 17841281393295360000

Offset: 0

N. J. A. Sloane, Jan 17 2000

Denominator of probability of a derangement of n things (A000166(n)/n!).

Also numerators of successive convergents to e using continued fraction 2 +1/(1 +1/(2 +2/(3 +3/(4 +4/(5 +5/(6 +6/(7 +7/8 +...))))))).

			1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

G. C. Greubel, Table of n, a(n) for n = 0..450 (terms 0..100 from T. D. Noe)
Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
Eric Weisstein's World of Mathematics, Subfactorial

Cf. A053557 (numerators), A053518-A053520. See also A103816.

a(n) = (D(n, n) of A103360), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360).

[Denominator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..20]]; // G. C. Greubel, May 16 2019

Table[Denominator[Sum[(-1)^k/k!, {k, 0, n}]], {n, 0, 20}] (* Robert G. Wilson v, Oct 13 2005 *)
Table[ Denominator[1 - Subfactorial[n]/n!], {n, 0, 22}] (* Jean-François Alcover, Feb 11 2014 *)
Denominator[Accumulate[Table[(-1)^k/k!,{k,0,30}]]] (* Harvey P. Dale, Aug 22 2016 *)

for(n=0,50, print1(denominator(sum(k=0,n,(-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017

from math import factorial
from fractions import Fraction
def A053556(n): return sum(Fraction(-1 if k&1 else 1,factorial(k)) for k in range(n+1)).denominator # Chai Wah Wu, Jul 31 2023

[denominator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..20)] # G. C. Greubel, May 16 2019

Let exp(-x)/(1-x) = Sum_{n>=0} (a_n/b_n) * x^n. Then sequence b_n is A053556. - Aleksandar Petojevic, Apr 14 2004

More terms from Vladeta Jovovic, Mar 31 2000

A000757 Number of cyclic permutations of [n] with no i -> i+1 (mod n).

Original entry on oeis.org

1, 0, 0, 1, 1, 8, 36, 229, 1625, 13208, 120288, 1214673, 13469897, 162744944, 2128047988, 29943053061, 451123462673, 7245940789072, 123604151490592, 2231697509543361, 42519034050101745, 852495597142800376

Offset: 0

N. J. A. Sloane

			a(4)=1 because from the 4!/4=6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) only (1,4,3,2) has no successor pair (i,i+1). Note that (4,1) is also a successor pair. - _Wolfdieter Lang_, Jan 21 2008
a(3) = 1 = 2! - 3*1! + 3*0! - 1. a(4) = 1 = 3! - 4*2! + 6*1! - 4*0! + 1. - _Michael Somos_, Mar 28 2011
G.f. = 1 + x^3 + x^4 + 8*x^5 + 36*x^6 + 229*x^7 + 1625*x^8 + 13208*x^9 + ...

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, pp. 182, 183, Table 5.6.
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).
R. P. Stanley, Space Programs Summary. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, Vol. 37-40-4 (1966), pp. 208-214.
R. P. Stanley, Enumerative Combinatorics I, Chap. 2, Exercise 8, p. 88.
N. Ya. Vilenkin, Combinatorics, pp. 56-57, Q_n = a(n), n >= 3. Academic Press, 1971. On the Merry Go-Round.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Roland Bacher, Counting Packings of Generic Subsets in Finite Groups, Electr. J. Combinatorics, 19 (2012), #P7. - _N. J. A. Sloane_, Feb 06 2013
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016.
S. M. Jacob, The enumeration of the Latin rectangle of depth three..., Proc. London Math. Soc., 31 (1928), 329-336.
R. Moreno and L. M. Rivera, Blocks in Cycles and k-commuting Permutations, arXiv preprint arXiv:1306.5708 [math.CO], 2013.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014.
R. P. Stanley, Permutations with no runs of length 2, Space Programs Summary. Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, Vol. 37-40-4 (1966), pp. 208-214. [Annotated scanned copy]

Cf. A000142, A000166, A002741.

a[n_] := (-1)^n + Sum[(-1)^k*n!/((n-k)*k!), {k, 0, n-1}]; Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Aug 30 2011, after Michael Somos *)

{a(n) = if( n<0, 0, (-1)^n + sum( k=0, n-1, (-1)^k * binomial( n, k) * (n - k - 1)!))}; /* Michael Somos, Jun 21 2002 */

From Michael Somos, Jun 21 2002: (Start)
a(n) = (-1)^n + Sum_{k=0..n-1} (-1)^k*binomial(n, k)*(n-k-1)!;
e.g.f.: (1 - log(1 - x)) / e^x;
a(n) = (n-3) * a(n-1) + (n-2) * (2*a(n-2) + a(n-3)).
a(n) = (n-2) * a(n-1) + (n-1) * a(n-2) - (-1)^n, if n > 0.
a(n) = (-1)^n + A002741(n). (End)
a(n) = n-th forward difference of [1, 1, 1, 2, 6, 24, ...] (factorials A000142 with 1 prepended). - Michael Somos, Mar 28 2011
a(n) = Sum_{j=3..n} (-1)^(n-j)*D(j-1), n >= 3, with the derangements numbers (subfactorials) D(n) = A000166(n).
a(n) + a(n+1) = A000166(n). - Aaron Meyerowitz, Feb 08 2014
a(n) ~ exp(-1)*(n-1)! * (1 - 1/n + 1/n^3 + 1/n^4 - 2/n^5 - 9/n^6 - 9/n^7 + 50/n^8 + 267/n^9 + 413/n^10 + ...), numerators are A000587. - Vaclav Kotesovec, Jul 03 2016

Better description from Len Smiley

Additional comments from Michael Somos, Jun 21 2002

A053557 Numerator of Sum_{k=0..n} (-1)^k/k!.

Original entry on oeis.org

1, 0, 1, 1, 3, 11, 53, 103, 2119, 16687, 16481, 1468457, 16019531, 63633137, 2467007773, 34361893981, 15549624751, 8178130767479, 138547156531409, 92079694567171, 4282366656425369, 72289643288657479, 6563440628747948887, 39299278806015611311

Offset: 0

N. J. A. Sloane, Jan 17 2000

Numerator of probability of a derangement of n things (A000166(n)/n! or !n/n!).

Also denominators of successive convergents to e using continued fraction 2 + 1/(1 + 1/(2 + 2/(3 + 3/(4 + 4/(5 + 5/(6 + 6/(7 + 7/(8 + ...)))))))).

			1, 0, 1/2, 1/3, 3/8, 11/30, 53/144, 103/280, 2119/5760, ...

L. Lorentzen and H. Waadeland, Continued Fractions with Applications, North-Holland 1992, p. 562.
E. Maor, e: The Story of a Number, Princeton Univ. Press 1994, pp. 151 and 157.

G. C. Greubel, Table of n, a(n) for n = 0..450 (terms 0..100 from T. D. Noe)
Leonhardo Eulero, Introductio in analysin infinitorum. Tomus primus, Lausanne, 1748.
L. Euler, Introduction à l'analyse infinitésimale, Tome premier, Tome second, trad. du latin en français par J. B. Labey, Paris, 1796-1797.
Eric Weisstein's World of Mathematics, Continued Fraction Constants
Eric Weisstein's World of Mathematics, Generalized Continued Fraction
Eric Weisstein's World of Mathematics, Subfactorial

Cf. A000166/A000142, A053556 (denominators), A053518-A053520. See also A103816.

a(n) = (N(n, n) of A103361), A053557/A053556 = A000166/n! = (N(n, n) of A103361)/(D(n, n) of A103360), Cf. A053518-A053520.

[Numerator( (&+[(-1)^k/Factorial(k): k in [0..n]]) ): n in [0..30]]; // G. C. Greubel, May 16 2019

Numerator[CoefficientList[Series[Exp[-x]/(1-x), {x, 0, 30}], x]] (* Jean-François Alcover, Nov 18 2011 *)
Table[Numerator[Sum[(-1)^k/k!,{k,0,n}]],{n,0,30}] (* Harvey P. Dale, Dec 02 2011 *)
Join[{1, 0}, Numerator[RecurrenceTable[{a[n]==a[n-1]+a[n-2]/(n-2), a[1] ==0, a[2]==1}, a, {n,2,30}]]] (* Terry D. Grant, May 07 2017; corrected by G. C. Greubel, May 16 2019 *)

for(n=0, 30, print1(numerator(sum(k=0,n, (-1)^k/k!)), ", ")) \\ G. C. Greubel, Nov 05 2017

from fractions import Fraction
from math import factorial
def A053557(n): return sum(Fraction(-1 if k&1 else 1,factorial(k)) for k in range(n+1)).numerator # Chai Wah Wu, Jul 31 2023

[numerator(sum((-1)^k/factorial(k) for k in (0..n))) for n in (0..30)] # G. C. Greubel, May 16 2019

Let exp(-x)/(1-x) = Sum_{n >= 0} (a_n/b_n)*x^n. Then sequence a_n is A053557. - Aleksandar Petojevic, Apr 14 2004

More terms from Vladeta Jovovic, Mar 31 2000

A001499 Number of n X n matrices with exactly 2 1's in each row and column, other entries 0.

Original entry on oeis.org

1, 0, 1, 6, 90, 2040, 67950, 3110940, 187530840, 14398171200, 1371785398200, 158815387962000, 21959547410077200, 3574340599104475200, 676508133623135814000, 147320988741542099484000, 36574751938491748341360000, 10268902998771351157327104000

Offset: 0

N. J. A. Sloane

Or, number of labeled 2-regular relations of order n.

Also number of ways to arrange 2n rooks on an n X n chessboard, with no more than 2 rooks in each row and column (no 3 in a line). - Vaclav Kotesovec, Aug 03 2013

R. Bricard, L'Intermédiaire des Mathématiciens, 8 (1901), 312-313.
L. Comtet, Advanced Combinatorics, Reidel, 1974, Sect. 6.3 Multipermutations, pp. 235-236, P(n,2), bipermutations.
L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review. Solution by D. E. Knuth. Reprinted in Problems in Applied Mathematics, ed. M. Klamkin, SIAM, 1990, p. 350.
Shanzhen Gao and Kenneth Matheis, Closed formulas and integer sequences arising from the enumeration of (0,1)-matrices with row sum two and some constant column sums. In Proceedings of the Forty-First Southeastern International Conference on Combinatorics, Graph Theory and Computing. Congr. Numer. 202 (2010), 45-53.
J. T. Lewis, Maximal L-free subsets of a squarefree array, Congressus Numerantium, 141 (1999), 151-155.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (b).
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.
J. H. van Lint and R. M. Wilson, A Course in Combinatorics (Cambridge University Press, Cambridge, 1992), pp. 152-153. [The second edition is said to be a better reference.]

Alois P. Heinz, Table of n, a(n) for n = 0..200 (terms n = 0..48 from R. W. Robinson)
H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.
Paul Barry, On the Connection Coefficients of the Chebyshev-Boubaker polynomials, The Scientific World Journal, Volume 2013 (2013), Article ID 657806.
L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782.
Sally Cockburn and Joshua Lesperance, Deranged Socks, Mathematics Magazine, Vol. 86, no. 2, April 2013, pp. 97-109.
L. Erlebach and O. Ruehr, Problem 79-5, SIAM Review, Vol. 21, No. 1 (Jan., 1979), p. 140. Solution by D. E. Knuth, SIAM Review, Vol. 22, No. 1 (Jan., 1980), pp. 101-102.
M. E. Kuczma, 0-1-Matrices with Line-Sums Equal to 2, Am. Math. Month. 99 (1992) 959-961, E3419.
Rui-Li Liu and Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Zhonghua Tan, Shanzhen Gao, and H. Niederhausen, Enumeration of (0,1) matrices with constant row and column sums, Appl. Math. Chin. Univ. 21 (4) (2006) 479-486.
Bo-Ying Wang and Fuzhen Zhang, On the precise number of (0,1)-matrices in A(R,S), Discrete Math. 187 (1998), no. 1-3, 211--220. MR1630720 (99f:05010). - From _N. J. A. Sloane_, Jun 07 2012
Index entries for sequences related to binary matrices

Cf. A000681, A053871, A123544 (connected relations), A000986 (symmetric matrices), A007107 (traceless matrices).
Cf. A000290, A002411, A005408.
Cf. A001501. Column 2 of A008300. Row sums of A284989.

a001499 n = a001499_list !! n
a001499_list = 1 : 0 : 1 : zipWith (*) (drop 2 a002411_list)
   (zipWith (+) (zipWith (*) [3, 5 ..] $ tail a001499_list)
                (zipWith (*) (tail a000290_list) a001499_list))
-- Reinhard Zumkeller, Jun 02 2013

a[n_] := (n-1)*n!*Gamma[n-1/2]*Hypergeometric1F1[2-n, 3/2-n, -1/2]/Sqrt[Pi]; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Oct 06 2011, after first formula *)

a(n)=if(n<2,n==0,(n^2-n)*(a(n-1)+(n-1)/2*a(n-2)))

seq(n)={Vec(serlaplace(serlaplace(exp(-x/2 + O(x*x^n))/sqrt(1-x + O(x*x^n)))))}; \\ Andrew Howroyd, Sep 09 2018

a(n) = (n! (n-1) Gamma(n-1/2) / Gamma(1/2) ) * 1F1[2-n; 3/2-n; -1/2] [Erlebach and Ruehr]. This representation is exact, asymptotic and convergent.
D-finite with recurrence 2*a(n) -2*n*(n-1)*a(n-1) -n*(n-1)^2*a(n-2)=0.
a(n) ~ 2 sqrt(Pi) n^(2n + 1/2) e^(-2n - 1/2) [Knuth]
a(n) = (1/2)*n*(n-1)^2 * ( (2*n-3)*a(n-2) + (n-2)^2*a(n-3) ) (from Anand et al.)
Sum_{n >= 0} a(n)*x^n/(n!)^2 = exp(-x/2)/sqrt(1-x); a(n) = n(n-1)/2 [ 2 a(n-1) + (n-1) a(n-2) ] (Bricard)
b_n = a_n/n! satisfies b_n = (n-1)(b_{n-1} + b_{n-2}/2); e.g.f. for {b_n} and for derangements (A000166) are related by D(x) = B(x)^2.
Limit_(n->infinity) sqrt(n)*a(n)/(n!)^2 = A096411 [Kuczma]. - R. J. Mathar, Sep 21 2007
a(n) = 4^(-n) * n!^2 * Sum_{i=0..n} (-2)^i * (2*n - 2*i)! / (i!*(n-i)!^2). - Shanzhen Gao, Feb 15 2010

cp's OEIS Frontend

A000186 Number of 3 X n Latin rectangles in which the first row is in order.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

SageMath

Formula

Extensions

A047920 Triangular array formed from successive differences of factorial numbers.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A055790 a(n) = n*a(n-1) + (n-2)*a(n-2), a(0) = 0, a(1) = 2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A000261 a(n) = n*a(n-1) + (n-3)*a(n-2), with a(1) = 0, a(2) = 1.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A000387 Rencontres numbers: number of permutations of [n] with exactly two fixed points.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Formula

Extensions

A008291 Triangle of rencontres numbers.

Views

Author

A055790 a(n) = na(n-1) + (n-2)a(n-2), a(0) = 0, a(1) = 2.

A000261 a(n) = na(n-1) + (n-3)a(n-2), with a(1) = 0, a(2) = 1.