A053871 -id:A053871 - cp's OEIS frontend

A155518 Number of permutations p of {1,2,...,n} such that p(j) + p(n+1-j) != n+1 for all j.

1, 0, 0, 4, 16, 64, 384, 2880, 23040, 208896, 2088960, 23193600, 278323200, 3640688640, 50969640960, 768126320640, 12290021130240, 209688566169600, 3774394191052800, 71921062285148160, 1438421245702963200

Offset: 0

Emeric Deutsch, Jan 26 2009

nonn

			a(3)=4 because we have 132, 312, 213 and 231 (123 and 321 do not qualify).

Cf. A053871, A155517, A155519.

g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: a := proc (n) if `mod`(n, 2) = 1 then factorial((1/2)*n-1/2)*2^((1/2)*n-1/2)*g[(1/2)*n+1/2] else factorial((1/2)*n)*2^((1/2)*n)*g[(1/2)*n] end if end proc: seq(a(n), n = 0 .. 24);

a(n) = A155517(n,0).

a(2n-1) = (n-1)!*2^(n-1)*g(n), a(2n) = n!*2^n*g(n), where g(n) = A053871(n) is defined by g(0)=1, g(1)=0, g(n) = 2(n-1)*(g(n-1) + g(n-2)) for n>=2.

A189849 a(0)=1, a(1)=0, a(n) = 4n(n-1)(a(n-1) + 2(n-1)*a(n-2)).

Original entry on oeis.org

1, 0, 16, 384, 23040, 2088960, 278323200, 50969640960, 12290021130240, 3774394191052800, 1438421245702963200, 666120016990568448000, 368420070161105761075200, 239869937154980747988172800, 181598769336835394381021184000, 158184707164826878472739618816000

Offset: 0

Stewart Herring, Apr 29 2011

The number of ways n couples can sit in rows of two seats with no person next to their partner.

a(n)/(2n)! gives the probability of this is and tends to exp(-1/2) as n tends to infinity.

G. C. Greubel, Table of n, a(n) for n = 0..224

Cf. A053871, A333706.

[(-2)^n*Factorial(n)*(&+[(-1/2)^k*Binomial(n,k)*Factorial(2*k)/Factorial(k): k in [0..n]]): n in [0..20]]; // G. C. Greubel, Jan 13 2018

a:= n-> (-2)^n*n!*add((-1/2)^i*binomial(n, i)*(2*i)!/i!, i=0..n): seq(a(n), n=0..20);

Table[(-2)^n*n!*Sum[(-1/2)^i*Binomial[n,i]*(2*i)!/i!,{i,0,n}],{n,1,20}]
RecurrenceTable[{a[0]==1,a[1]==0,a[n]==4n(n-1)(a[n-1]+2(n-1)a[n-2])},a,{n,20}] (* Harvey P. Dale, May 02 2012 *)

a[0]:1$ a[1]:0$ a[n]:=4*n*(n-1)*(a[n-1]+2*(n-1)*a[n-2])$ makelist(a[n], n, 0, 13);  /* Bruno Berselli, May 23 2011 */

for(n=0,30, print1((-2)^n*n!*sum(k=0,n, (-1/2)^k*binomial(n,k)*(2*k)!/k!), ", ")) \\ G. C. Greubel, Jan 13 2018

a(n) = (-2)^n*n!*hypergeom([ -n, 1/2],[],2).
a(n) = (n!)^2 times the coefficient of x^n in the expansion of exp(-2*x)/sqrt(1-4*x).
a(n) = 2^n*n!*A053871(n).
a(n) = A333706(2n,n). - Alois P. Heinz, Apr 10 2020

A292080 Number of nonequivalent ways to place n non-attacking rooks on an n X n board with no rook on 2 main diagonals up to rotations and reflections of the board.

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 14, 84, 630, 6096, 55336, 672160, 7409300, 104999520, 1366363752, 22068387264, 331233939624, 6005919062528, 102144359744192, 2054811316442112, 39053339674065360, 863259240785840640, 18132529836143846560, 436899062862222484480

Offset: 0

Andrew Howroyd, Sep 12 2017

nonn

For odd n, there are no symmetrical configurations of non-attacking rooks without a rook in the main diagonal, so a(2n+1) = A003471(2n+1) / 8. For even n, configurations with rotational and diagonal symmetry are possible.

			Case n=4: The 2 nonequivalent solutions are:
   _ x _ _     _ x _ _
   x _ _ _     _ _ _ x
   _ _ _ x     x _ _ _
   _ _ x _     _ _ x _
Case n=5: The 2 nonequivalent solutions are:
   _ x _ _ _   _ x _ _ _
   x _ _ _ _   _ _ _ _ x
   _ _ _ x _   x _ _ _ _
   _ _ _ _ x   _ _ x _ _
   _ _ x _ _   _ _ _ x _

Andrew Howroyd, Table of n, a(n) for n = 0..100
Andrew Howroyd, Nonequivalent and Symmetric Solutions

Cf. A000166, A000903, A003471, A037224, A053871, A064280.

sf[n_] := n! * SeriesCoefficient[Exp[-x ] / (1 - x), {x, 0, n}];
F[n_] := (Clear[v]; v[_] = 0; For[m = 4, m <= n, m++, v[m] = (m - 1)*v[m - 1] + 2*If[OddQ[m], (m - 1)*v[m - 2], (m - 2)*If[m == 4, 1, v[m - 4]]]]; v[n]);
d[n_] := Sum[(-1)^(n-k)*Binomial[n, k]*(2k)!/(2^k*k!), {k, 0, n}];
R[n_] := If[OddQ[n], 0, (n - 1)!*2/(n/2 - 1)!];
a[0] = 1; a[n_] := (F[n] + If[OddQ[n], 0, m = n/2; 2^m * sf[m] + 2*R[m] + 2*d[m]])/8;
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 28 2017, after Andrew Howroyd *)

\\ here sf is A000166, F is A003471, D is A053871, R(n) is A037224(2n).
sf(n) = {n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}
F(n) = {my(v = vector(n)); for(n=4,length(v),v[n]=(n-1)*v[n-1]+2*if(n%2==1,(n-1)*v[n-2],(n-2)*if(n==4,1,v[n-4]))); v[n]}
D(n) = {sum(k=0, n, (-1)^(n-k) * binomial(n,k) * (2*k)!/(2^k*k!))}
R(n) = {if(n%2==1, 0, (n-1)!*2/(n/2-1)!)}
a(n) = {(F(n) + if(n%2==1, 0, my(m=n/2); 2^m * sf(m) + 2*R(m) + 2*D(m)))/8}

a(2n+1) = A003471(2n+1) / 8, a(2n) = (A003471(2n) + 2^n * A000166(n) + 2*A037224(2*n) + 2*A053871(n)) / 8.

A370253 Number of deranged matchings of 2n people with partners (of either sex) such that at least one person is matched with their spouse.

Original entry on oeis.org

0, 1, 1, 7, 45, 401, 4355, 56127, 836353, 14144545, 267629139, 5601014255, 128455425593, 3203605245777, 86317343312395, 2498680706048191, 77336483434140705, 2548534969132415297, 89087730603300393443, 3292572900736818264015, 128281460895447809211529

Offset: 0

Sam Coutteau, Feb 13 2024

nonn

			For n=0, there is no matching which has at least one person matched with their original partner.
For n=1, there are only 2 people, so there is only one way to match them and it is with their original partner.
For n=2, we have two couples, A0 with A1, and B0 with B1. Of the three ways to match them [(A0,A1),(B0,B1)], [(A0,B0),(A1,B1)] and [(A0,B1),(A1,B0)], only the first matching has a person matched up with their original partner.

Alois P. Heinz, Table of n, a(n) for n = 0..404

Cf. A001147 (total number of matchings for 2n people).
Cf. A053871 (number of deranged matchings of 2n people with partners (of either sex) other than their spouse).
Cf. A055140, A057427.

a:= proc(n) option remember; `if`(n<3, signum(n),
      (4*n-7)*a(n-1)-2*(2*n^2-10*n+11)*a(n-2)-2*(n-2)*(2*n-5)*a(n-3))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Feb 14 2024

a[n_] := Sum[(-1)^(n-i+1)*Binomial[n, i]*(2i-1)!!, {i, 0, n-1}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 29 2024 *)

import math
A001147 = lambda i: math.factorial(2*i) // ( 2 ** i * math.factorial(i) )
A370253 = lambda n: int( sum( (-1)**(i+1) * math.comb(n,n-i) * A001147(n-i) for i in range(1,n+1) ) )
print( ", ".join( str(A370253(i)) for i in range(0,21) ) )

a(n) = A001147(n) - A053871(n).
a(n) = Sum_{i=0..n-1} (-1)^(n - i + 1) * binomial(n,i)*A001147(i).
a(n) mod 2 = A057427(n).
a(n) = Sum_{k=1..n} A055140(n,k). - Alois P. Heinz, Feb 14 2024

A297474 Number of maximal matchings in the n-cocktail party graph.

Original entry on oeis.org

1, 2, 14, 92, 844, 9304, 121288, 1822736, 31030928, 590248736, 12406395616, 285558273472, 7143371664064, 192972180052352, 5598713198048384, 173627942889668864, 5731684010612723968, 200669613102747214336, 7426773564495661485568, 289713958515451427511296

Offset: 1

Eric W. Weisstein, Dec 30 2017

nonn

A maximal matching in the n-cocktail party graph is either a perfect matching or a matching with a single unmatched pair. - Andrew Howroyd, Dec 30 2017

Eric Weisstein's World of Mathematics, Cocktail Party Graph
Eric Weisstein's World of Mathematics, Matching
Eric Weisstein's World of Mathematics, Maximal Independent Edge Set

Cf. A053871.

Table[(-1)^(n + 1) (n HypergeometricPFQ[{1/2, 1 - n}, {}, 2] - HypergeometricPFQ[{1/2, -n}, {}, 2]), {n, 20}]
Table[-I (-1)^n (n HypergeometricU[1/2, n + 1/2, -1/2] - HypergeometricU[1/2, n + 3/2, -1/2])/Sqrt[2], {n, 20}]

\\ here b(n) is A053871.
b(n)={if(n<1, n==0, sum(k=0, n, (-1)^(n-k)*binomial(n,k)*(2*k)!/(2^k*k!)))}
a(n)=b(n) + n*b(n-1); \\ Andrew Howroyd, Dec 30 2017

a(n) = A053871(n) + n*A053871(n-1). - Andrew Howroyd, Dec 30 2017

a(9)-a(20) from Andrew Howroyd, Dec 30 2017

A382134 Number of completely asymmetric matchings (not containing centered or coupled arcs) of [2n].

Original entry on oeis.org

1, 0, 0, 8, 48, 384, 4480, 59520, 897792, 15368192, 293769216, 6198589440, 143130972160, 3590253477888, 97214510235648, 2826205634330624, 87801981951344640, 2902989352269250560, 101776549707306237952, 3771425415371470405632, 147285455218020210180096

Offset: 0

R. J. Mathar, Mar 17 2025

nonn

Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, theorem 5.1

Cf. A000079, A001205, A047974, A053871.

g:= exp(-x-x^2)/sqrt(1-2*x) ;
seq( coeftayl(g,x=0,n)*n!,n=0..10) ;

E.g.f: exp(-x-x^2)/sqrt(1-2*x).
a(n) = 2^n * A001205(n).
D-finite with recurrence a(n) +2*(-n+1)*a(n-1) -4*(n-1)*(n-2)*a(n-3)=0.

A372260 Triangle read by rows: T(n, k) = (T(n-1, k-1) + T(n-1, k)) * 2 * n with initial values T(n, 0) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A001147(i) and T(i, j) = 0 if j > i.

Original entry on oeis.org

1, 0, 2, 2, 8, 8, 8, 60, 96, 48, 60, 544, 1248, 1152, 384, 544, 6040, 17920, 24000, 15360, 3840, 6040, 79008, 287520, 503040, 472320, 230400, 46080, 79008, 1190672, 5131392, 11067840, 13655040, 9838080, 3870720, 645120, 1190672, 20314880, 101153024, 259187712, 395566080, 375889920, 219340800, 72253440, 10321920

Offset: 0

Werner Schulte, Apr 24 2024

			Triangle T(n, k) starts:
n\k :     0       1       2        3        4       5       6      7
====================================================================
  0 :     1
  1 :     0       2
  2 :     2       8       8
  3 :     8      60      96       48
  4 :    60     544    1248     1152      384
  5 :   544    6040   17920    24000    15360    3840
  6 :  6040   79008  287520   503040   472320  230400   46080
  7 : 79008 1190672 5131392 11067840 13655040 9838080 3870720 645120
  etc.

Cf. A053871 (column 0), 2*A179540 (column 1), A000165 (main diagonal).

Cf. A001147.

T := proc(n, k) option remember; `if`(k > n, 0, `if`(k = n, 2^n * n!, `if`(k = 0, `if`(n < 2, 1 - n, (2*n - 2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n))) end:
for n from 0 to 7 do seq(T(n, k), k = 0..n) od;  # Peter Luschny, Apr 25 2024

T[n_,k_]:=n!SeriesCoefficient[(Exp[-t]/Sqrt[1 - 2*t])*(2*t/(1-2*t))^k,{t,0,n}]; Table[T[n,k],{n,0,8},{k,0,n}]//Flatten (* Stefano Spezia, Apr 25 2024 *)

{  T(n, k) = if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1-n,
         (2*n-2) * (T(n-1, k) + T(n-2, k))), (T(n-1, k-1) + T(n-1, k)) * 2*n)))  }

memo = Map(); memoize(f, A[..]) =
{ my(res);
  if(!mapisdefined(memo, [f, A], &res), res = call(f, A);
  mapput(memo, [f, A], res)); res; }
T(n, k) =
{ if(k>n, 0, if(k==n, 2^n * n!, if(k==0, if(n<2, 1 - n,
  (2 * n - 2) * (memoize(T, n-1, k) + memoize(T, n-2, k))),
  (memoize(T, n-1, k-1) + memoize(T, n-1, k)) * 2 * n))); }

T(n, 0) = (2*n - 2) * (T(n-1, 0) + T(n-2, 0)) for n > 1 with initial values T(n, 0) = 1 - n for n < 2 (see A053871).
T(n, k) = (Sum_{i=0..k} binomial(k, i) * T(n-i, 0)) * 2^(2*k) * binomial(n, k) / binomial(2*k, k).
E.g.f. of column k: (exp(-t) / sqrt(1 - 2*t)) * (2*t / (1 - 2*t))^k.
E.g.f.: exp((2*x / (1 - 2*t) - 1) * t) / sqrt(1 - 2*t).

A382139 Number of matchings of [2n] with no coupled arcs.

Original entry on oeis.org

1, 1, 1, 9, 81, 705, 7665, 100905, 1524705, 26022465, 496042785, 10445342985, 240779831985, 6030718158465, 163087008669585, 4735950860666025, 146987669673669825, 4855606200012593025, 170101350767940617025, 6298861062893921346825, 245834199405298416337425

Offset: 0

R. J. Mathar, Mar 17 2025

nonn

Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, eq (17) at r=1, s=2.

Cf. A047974, A053871, A382134, A067994, A001147.

g:= exp(-x^2)/sqrt(1-2*x) ;
seq( coeftayl(g,x=0,n)*n!,n=0..10) ;

E.g.f: exp(-x^2)/sqrt(1-2*x).
Exponential convolution of A067994 and A001147.
D-finite with recurrence a(n) +(-2*n+1)*a(n-1) +2*(n-1)*a(n-2) -4*(n-1)*(n-2)*a(n-3)=0.

cp's OEIS Frontend

A155518 Number of permutations p of {1,2,...,n} such that p(j) + p(n+1-j) != n+1 for all j.

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A189849 a(0)=1, a(1)=0, a(n) = 4*n*(n-1)*(a(n-1) + 2*(n-1)*a(n-2)).

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A292080 Number of nonequivalent ways to place n non-attacking rooks on an n X n board with no rook on 2 main diagonals up to rotations and reflections of the board.

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A370253 Number of deranged matchings of 2n people with partners (of either sex) such that at least one person is matched with their spouse.

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A297474 Number of maximal matchings in the n-cocktail party graph.

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Extensions

A382134 Number of completely asymmetric matchings (not containing centered or coupled arcs) of [2n].

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A372260 Triangle read by rows: T(n, k) = (T(n-1, k-1) + T(n-1, k)) * 2 * n with initial values T(n, 0) = Sum_{i=0..n} (-1)^(n-i) * binomial(n, i) * A001147(i) and T(i, j) = 0 if j > i.

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A189849 a(0)=1, a(1)=0, a(n) = 4n(n-1)(a(n-1) + 2(n-1)*a(n-2)).