A000271 -id:A000271 - cp's OEIS frontend

A127548 O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.

1, 1, 0, 1, 4, 19, 112, 771, 6088, 54213, 537392, 5867925, 69975308, 904788263, 12607819040, 188341689287, 3002539594128, 50878366664393, 913161208490016, 17304836525709097, 345279674107957524, 7235298537356113339

Offset: 0

Vladeta Jovovic, Jun 27 2007

a(n+1) = inverse binomial transform of A013999 = Sum_{k=0..n} binomial(n,k)*(-1)^(n-k)*A013999(k). - Emanuele Munarini, Jul 01 2013

Seiichi Manyama, Table of n, a(n) for n = 0..450

Cf. A000179, A078480, A013999, A000271, A000904, A078509.

A127548 := proc(n) if n = 0 then 1 ; else add(factorial(s)*(-1)^(n-s)*binomial(s+n-1,2*s-1),s=1..n) ; fi ; end: for n from 0 to 20 do printf("%d,",A127548(n)) ; od ; # R. J. Mathar, Jul 13 2007

nn = 21; CoefficientList[Series[Sum[n!*(x/(1 + x)^2)^n, {n, 0, nn}], {x, 0, nn}], x] (* Michael De Vlieger, Sep 04 2016 *)

import math
def binomial(n,m):
    a=1
    for k in range(n-m+1,n+1):
        a *= k
    return a//math.factorial(m)
def A127548(n):
    if n == 0:
        return 1
    a=0
    for s in range(1,n+1):
        a += (-1)**(n-s)*binomial(s+n-1,2*s-1)*math.factorial(s)
    return a
for n in range(30):
    print(A127548(n))
# R. J. Mathar, Oct 20 2009

a(n) = Sum_{s=1..n} (-1)^(n-s)*s!*C(s+n-1,2s-1) if n>=1, where C(a,b)=binomial(a,b). - R. J. Mathar, Jul 13 2007
G.f.: Q(0) where Q(k) = 1 + (2*k + 1)*x/( (1+x)^2- 2*x*(1+x)^2*(k+1)/(2*x*(k+1) + (1+x)^2/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Mar 08 2013
a(n) = A000271(n) + A000271(n-1). - Peter Bala, Sep 02 2016
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Oct 31 2017

More terms from R. J. Mathar, Jul 13 2007

More terms from R. J. Mathar, Oct 20 2009

A137886 Number of (directed) Hamiltonian paths in the n-crown graph.

Original entry on oeis.org

12, 144, 3840, 138240, 6804000, 436504320, 35417088000, 3546005299200, 429451518988800, 61883150757120000, 10463789706751180800, 2051763183437532364800, 461802751261297205760000, 118254166096501129863168000

Offset: 3

Eric W. Weisstein, Feb 20 2008

nonn

The reference to A094047 arises in the formula because that sequence is also the number of directed Hamiltonian cycles in the n-crown graph. (Each cycle can be broken in 2n ways to give a path.) - Andrew Howroyd, Feb 21 2016

Also, the number of ways of seating n married couples at 2*n chairs arranged side-by-side in a straight line, men and women in alternate positions, so that no husband is next to his wife. - Andrew Howroyd, Sep 19 2017

Seiichi Manyama, Table of n, a(n) for n = 3..253 (terms 3..50 from Andrew Howroyd)
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems
Eric Weisstein's World of Mathematics, Crown Graph
Eric Weisstein's World of Mathematics, Hamiltonian Path

Cf. A000271, A059375, A094047, A259212, A270174.

Table[2 n! Sum[(-1)^(n - k) k! Binomial[n + k, 2 k], {k, 0, n}], {n, 3, 20}] (* Eric W. Weisstein, Sep 20 2017 *)
Table[2 (-1)^n n! HypergeometricPFQ[{1, -n, n + 1}, {1/2}, 1/4], {n, 3, 20}] (* Eric W. Weisstein, Sep 20 2017 *)

/* needs the routine nhp() from the Alekseyev link */
{ A137886(n) = nhp( matrix(2*n,2*n,i,j, if(min(i,j)<=n && max(i,j)>n && abs(j-i)!=n, 1, 0)) ) }

For n>3, a(n) = 2*n*A094047(n) + n*a(n-1) = A059375(n) + n*a(n-1). - Andrew Howroyd, Feb 21 2016
a(n) ~ 4*Pi*n^(2*n+1) / exp(2*n+2). - Vaclav Kotesovec, Feb 25 2016
a(n) = (n-1)*n*a(n-1) + (n-1)^2*n*a(n-2) + (n-2)*(n-1)*n*a(n-3). - Vaclav Kotesovec, Feb 25 2016
a(n) = 2*n! * A000271(n). - Andrew Howroyd, Sep 19 2017

More terms from Max Alekseyev, Feb 13 2009
a(14) from Eric W. Weisstein, Jan 15 2014
a(15)-a(16) from Andrew Howroyd, Feb 21 2016

A273596 For n >= 2, a(n) is the number of slim rectangular diagrams of length n.

Original entry on oeis.org

1, 3, 9, 32, 139, 729, 4515, 32336, 263205, 2401183, 24275037, 269426592, 3257394143, 42615550453, 599875100487, 9040742057760, 145251748024649, 2478320458476795, 44755020000606961, 852823700470009056, 17101229029400788083, 359978633317886558801, 7936631162022905081707

Offset: 2

Tamas Dekany, May 26 2016

			The initial term is the diagram of the four element diamond shape lattice.

Vaclav Kotesovec, Table of n, a(n) for n = 2..400
P. Bala, Notes on A273596
Gábor Czédli, Tamás Dékány, Gergő Gyenizse, and Júlia Kulin, The number of slim rectangular lattices, Algebra Universalis, 2016, Volume 75, Issue 1, pp 33-50.

Cf. A273988, A000522, A143409, A000179, A000271, A000904, A127548.

Partial sums of A155857.

A273596 := proc (n) option remember; `if`(n = 2, 1, `if`(n = 3, 3, (n-2)*procname(n-1) + procname(n-2) + 2)) end: seq(A273596(n), n = 2..20); # Peter Bala, Jan 08 2017

x = 15;
SRectD = Table[0, {x}];
For[n = 2, n < x, n++,
For[a = 1, a < n, a++,
   For[b = 1, b <= n - a, b++,
    SRectD[[n]] +=
      Binomial[n - a - 1, b - 1]*
       Binomial[n - b - 1, a - 1]*(n - a - b)!;
    ]
   ]
  Print[n, " ", SRectD[[n]]]
]
(* Alternatively: *)
T[n_, k_] := HypergeometricPFQ[{k+1, k-n}, {}, -1];
Table[Sum[T[n,k], {k,0,n}], {n,0,22}] (* Peter Luschny, Oct 05 2017 *)

a(n)= sum(rps=1, n, sum(r=1, n, s = rps-r; binomial(n-r-1, s-1) * binomial(n-s-1, r-1) * (n-r-s)!)); \\ Michel Marcus, Jun 12 2016

a(n) = Sum_{1<=r,s; r+s<=n} binomial(n-r-1, s-1) * binomial(n-s-1, r-1) * (n-r-s)!.
a(n) ~ exp(2) * n! / n^2. - Vaclav Kotesovec, Jun 29 2016
a(n) = Sum_{k=0..n} hypergeom([k+1, k-n], [], -1). - Peter Luschny, Oct 05 2017
From Peter Bala, Jan 08 2018: (Start)
a(n) = Sum_{k = 0..n-2} k!*binomial(n+k-1, 2*k+1).
a(n) = (n - 2)*a(n-1) + a(n-2) + 2, with a(2) = 1, a(3) = 3.
a(n+2) = 1/n!*Sum_{k = 0..n} (-1)^(n-k)*binomial(n,k)* A000522(n)^2.
Row sums of array A143409 read as a triangle.
O.g.f.: Sum_{n >= 0} n!*x^(n+2)/(1 - x)^(2*n+2). Cf. A000179, A000271, A000904 and A127548.
O.g.f. with offset 0: 1/(1 - x) o 1/(1 - x) = 1 + 3*x + 9*x^2 + 32*x^3 + ..., where o denotes the white diamond multiplication of power series. See the Bala link for details. (End)

A292574 Number of permutations p of {1,2,...,n} such that p(i)-i not in {-1,0,1,2}.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 58, 499, 4814, 50284, 572228, 7050770, 93637691, 1334156612, 20308818956, 329025006637, 5653813150732, 102722614426328, 1967763318700136, 39640921470181124, 837836538203311613, 18539041315706787978, 428620090892592760870

Offset: 0

Andrew Howroyd, Sep 19 2017

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..40
D. Zeilberger, Automatic Enumeration of Generalized Ménage Numbers
D. Zeilberger, Automatic Enumeration of Generalized Menage Numbers, arXiv preprint arXiv:1401.1089 [math.CO], 2014.

Cf. A000271, A001883, A270174.

A097625 a(n) = Sum_{k=0..n} (-2)^k * binomial(2n-k,k) * (n-k)!.

Original entry on oeis.org

-1, 0, 2, -4, 8, 0, 80, 544, 5248, 53504, 601344, 7339520, 96797696, 1371889664, 20797212672, 335835828224, 5755617771520, 104346351861760, 1995288143593472, 40135085601325056, 847203499270995968

Offset: 1

Ralf Stephan, Sep 20 2004

sign

Permanent of certain n X 2 Toeplitz-(1,-1) matrices.

A. R. Kräuter, Über die Permanente gewisser zirkulärer Matrizen..., Sem. Loth. Combin. B11b (1984) 82-94

Cf. A000271.

A097625 := proc(n) add((-2)^k*(n-k)!*binomial(2*n-k,k),k=0..n) ; end proc:
seq(A097625(n),n=1..30) ; # R. J. Mathar, Sep 18 2011

A156368 A ménage triangle.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 3, 1, 3, 8, 6, 6, 1, 16, 35, 38, 20, 10, 1, 96, 211, 213, 134, 50, 15, 1, 675, 1459, 1479, 915, 385, 105, 21, 1, 5413, 11584, 11692, 7324, 3130, 952, 196, 28, 1, 48800, 103605, 104364, 65784, 28764, 9090, 2100, 336, 36, 1

Offset: 0

Paul Barry, Feb 08 2009

			Triangle begins:
   1;
   0,   1;
   0,   1,   1;
   1,   1,   3,   1;
   3,   8,   6,   6,  1;
  16,  35,  38,  20, 10,  1;
  96, 211, 213, 134, 50, 15,  1;

A. Kaufmann, Introduction à la combinatorique en vue des applications, p.188-189, Dunod, Paris, 1968. - Philippe Deléham, Apr 04 2014

Alois P. Heinz, Rows n = 0..140, flattened

Cf. A000142, A000271, A007318, A155856.

T[n_,k_]:= Sum[(-1)^(k+j)*Binomial[j, k]*Binomial[2*n-j, j]*(n-j)!, {j,0,n}];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 05 2021 *)

def A156368(n,k): return sum( (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*factorial(n-j) for j in (0..n) )
flatten([[A156368(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 05 2021

T(n, k) = Sum_{j=0..n} (-1)^(k+j)*binomial(j, k)*binomial(2*n-j, j)*(n-j)!.
T(n, 0) = A000271(n).
Sum_{k=0..n} T(n, k) = n!.
Equals A155856*A007318^{-1}.
G.f.: 1/(1 +x -x*y -x/(1 +x -x*y -x/(1 +x -x*y -2*x/(1 +x -x*y -2*x/(1 +x -x*y -3*x/(1 +x -x*y -3*x/(1 +x -x*y -4*x/(1 + ... (continued fraction).
G.f.: Sum_{n>=0} n! * x^n/(1 + (1-y)*x)^(2*n+1). - Ira M. Gessel, Jan 15 2013

A293042 Number of even permutations p of {1,...,n} such that p(i) is not i or i+1.

Original entry on oeis.org

1, 0, 0, 1, 1, 9, 47, 339, 2705, 24402, 244294, 2689669, 32297985, 420096147, 5883813373, 88287031271, 1412982765793, 24026200566404, 432554403678604, 8219863859175945, 164419973194802817, 3453229295483253853, 75978854506098365995, 1747670263607990439483

Offset: 0

Eric M. Schmidt, Sep 28 2017

nonn

Dmitry Efimov, Determinants of generalized binary band matrices, arXiv:1702.05655 [math.RA], 2017.

Cf. A000271, A293043.

a[n_] := (Sum[(-1)^(n-k) k! Binomial[n+k, 2 k], {k, 0, n}] + (-1)^(n-1)* Floor[(n-1)/2])/2; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 18 2019 *)

a(n) = (A000271(n) + (-1)^(n-1) * floor((n-1)/2))/2.

A293043 Number of odd permutations p of {1,...,n} such that p(i) is not i or i+1.

Original entry on oeis.org

0, 0, 0, 0, 2, 7, 49, 336, 2708, 24398, 244298, 2689664, 32297990, 420096141, 5883813379, 88287031264, 1412982765800, 24026200566396, 432554403678612, 8219863859175936, 164419973194802826, 3453229295483253843, 75978854506098366005, 1747670263607990439472

Offset: 0

Eric M. Schmidt, Sep 28 2017

nonn

Dmitry Efimov, Determinants of generalized binary band matrices, arXiv:1702.05655 [math.RA], 2017.

Cf. A000271, A293042.

a[n_] := (Sum[(-1)^(n-k) k! Binomial[n+k, 2 k], {k, 0, n}] + (-1)^n* Floor[(n - 1)/2])/2; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Feb 18 2019 *)

a(n) = (A000271(n) + (-1)^n * floor((n-1)/2))/2.

cp's OEIS Frontend

A127548 O.g.f.: Sum_{n>=0} n!*(x/(1+x)^2)^n.

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A137886 Number of (directed) Hamiltonian paths in the n-crown graph.

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PARI

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A273596 For n >= 2, a(n) is the number of slim rectangular diagrams of length n.

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A292574 Number of permutations p of {1,2,...,n} such that p(i)-i not in {-1,0,1,2}.

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A097625 a(n) = Sum_{k=0..n} (-2)^k * binomial(2n-k,k) * (n-k)!.

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A156368 A ménage triangle.

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A293042 Number of even permutations p of {1,...,n} such that p(i) is not i or i+1.

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A293043 Number of odd permutations p of {1,...,n} such that p(i) is not i or i+1.

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