A000255 -id:A000255 - cp's OEIS frontend

A090012 Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.

3, 9, 39, 213, 1395, 10617, 91911, 890901, 9552387, 112203465, 1432413063, 19743404469, 292164206259, 4619383947513, 77708277841575, 1385712098571957, 26108441941918851, 518231790473609481, 10808479322484810087

Offset: 1

Jaap Spies, Dec 13 2003

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

Indranil Ghosh, Table of n, a(n) for n = 1..447
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), pp. 197-210.

Cf. A000255, A000153, A000261, A001909, A001910, A090010, A055790, A090013-A090016.

A090012 := proc(n,d) local r; if (n=1) then r := d+1 elif (n=2) then r := (d+1)^2 else r := (n+d-1)*A090012(n-1,d)+(n-2)*A090012(n-2,d) fi; RETURN(r); end: seq(A090012(n,2),n=1..20);

t={3,9};Do[AppendTo[t,(n+1)*t[[-1]]+(n-2)*t[[-2]]],{n,3,19}];t (* Indranil Ghosh, Feb 21 2017 *)
RecurrenceTable[{a[1]==3,a[2]==9,a[n]==(n+1)a[n-1]+(n-2)a[n-2]},a,{n,20}] (* Harvey P. Dale, Sep 21 2017 *)

# Program to generate the b-file
print("1 3")
print("2 9")
a=3
b=9
c=(3+1)*b+(3-2)*a
for i in range(4, 40):
    print(str(i - 1)+" "+str(c))
    a=b
    b=c
    c=(i+1)*b+(i-2)*a # Indranil Ghosh, Feb 21 2017

a(n) = (n+1)*a(n-1) + (n-2)*a(n-2), a(1)=3, a(2)=9.
a(n) = A000153(n-1) + A000153(n), a(1)=3.
G.f.: W(0)/x -1/x, where W(k) = 1 - x*(k+3)/( x*(k+2) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 25 2013
a(n) ~ exp(-1) * n! * n^2 / 2. - Vaclav Kotesovec, Nov 30 2017

Corrected by Jaap Spies, Jan 26 2004

A090016 Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.

Original entry on oeis.org

7, 49, 399, 3689, 38087, 433713, 5394991, 72737161, 1056085191, 16423175153, 272275569167, 4792916427369, 89267526953479, 1753598009244529, 36232438035285807, 785431570870425353, 17822981129678644871

Offset: 1

Jaap Spies, Dec 13 2003

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

Indranil Ghosh, Table of n, a(n) for n = 1..444
Seok-Zun Song et al., Extremes of permanents of (0,1)-matrices, Lin. Algebra and its Applic. 373 (2003), pp. 197-210.

a(n) = A090010(n-1) + A090010(n), a(1)=7

Cf. A000255, A000153, A000261, A001909, A001910, A090010, A055790, A090012-A090015.

t={7,49};Do[AppendTo[t,(n+5)*t[[-1]]+(n-2)*t[[-2]]],{n,3,17}];t (* Indranil Ghosh, Feb 21 2017 *)

a(n) = (n+5)*a(n-1) + (n-2)*a(n-2), a(1)=7, a(2)=49
E.g.f.: 7*exp(-x)/(1-x)^8. - Vladeta Jovovic, Mar 19 2004
a(n) = (A000166(n-1)+7*A000166(n)+21*A000166(n+1)+35*A000166(n+2)+35*A000166(n+3)+21*A000166(n+4)+7*A000166(n+5)+A000166(n+6))/6!. - Vladeta Jovovic, Mar 19 2004
a(n) ~ exp(-1) * n! * n^6 / 6!. - Vaclav Kotesovec, Nov 30 2017

Corrected by Jaap Spies, Jan 26 2004

A153229 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

Original entry on oeis.org

0, 1, 0, 2, 4, 20, 100, 620, 4420, 35900, 326980, 3301820, 36614980, 442386620, 5784634180, 81393657020, 1226280710980, 19696509177020, 335990918918980, 6066382786809020, 115578717622022980, 2317323290554617020, 48773618881154822980, 1075227108896452857020

Offset: 0

Shaojun Ying (dolphinysj(AT)gmail.com), Dec 21 2008

Previous name was: Weighted Fibonacci numbers.
From Peter Bala, Aug 18 2013: (Start)
The sequence occurs in the evaluation of the integral I(n) := Integral_{u >= 0} exp(-u)*u^n/(1 + u) du.
The result is I(n) = A153229(n) + (-1)^n*I(0), where I(0) = Integral_{u >= 0} exp(-u)/(1 + u) du = 0.5963473623... is known as Gompertz's constant. See A073003.
Note also that I(n) = n!*Integral_{u >= 0} exp(-u)/(1 + u)^(n+1) du. (End)
((-1)^(n+1))*a(n) = p(n,-1), where the polynomials p are defined at A248664. - Clark Kimberling, Oct 11 2014

			a(20) = 19 * a(18) + 18 * a(19) = 19 * 335990918918980 + 18 * 6066382786809020 = 6383827459460620 + 109194890162562360 = 115578717622022980

Alois P. Heinz, Table of n, a(n) for n = 0..200

Cf. A000045, A000255, A000587, A058006.
First differences of A136580.
Column k=0 of A303697 (for n>0).

unsigned long a(unsigned int n) {
if (n == 0) return 0;
if (n == 1) return 1;
return (n - 1) * a(n - 2) + (n - 2) * a(n - 1); }

t1 := sum(n!*x^n, n=0..100): F := series(t1/(1+x), x, 100): for i from 0 to 40 do printf(`%d, `, i!-coeff(F, x, i)) od: # Zerinvary Lajos, Mar 22 2009
# second Maple program:
a:= proc(n) a(n):= `if`(n<2, n, (n-1)*a(n-2) +(n-2)*a(n-1)) end:
seq(a(n), n=0..25); # Alois P. Heinz, May 24 2013

Join[{a = 0}, Table[b = n! - a; a = b, {n, 0, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jun 28 2011 *)
RecurrenceTable[{a[0]==0,a[1]==1,a[n]==(n-1)a[n-2]+(n-2)a[n-1]},a,{n,30}] (* Harvey P. Dale, May 01 2020 *)

a(n)=if(n,my(t=(-1)^n);-t-sum(i=1,n-1,t*=-i),0); \\ Charles R Greathouse IV, Jun 28 2011

a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).
For n>=1, a(n) = A058006(n-1) * (-1)^(n-1).
G.f.: G(0)*x/(1+x)/2, where G(k)= 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 24 2013
G.f.: 2*x/(1+x)/G(0), where G(k)= 1 + 1/(1 - 1/(1 - 1/(2*x*(k+1)) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 29 2013
G.f.: W(0)*x/(1+sqrt(x))/(1+x), where W(k) = 1 + sqrt(x)/( 1 - sqrt(x)*(k+1)/(sqrt(x)*(k+1) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 17 2013
a(n) ~ (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), where numerators are Rao Uppuluri-Carpenter numbers, see A000587. - Vaclav Kotesovec, Mar 16 2015
E.g.f.: exp(1)/exp(x)*(Ei(1, 1-x)-Ei(1, 1)). - Alois P. Heinz, Jul 05 2018
a(n) = Sum_{k = 0..n-1} (-1)^(n-k-1) * k!. - Peter Bala, Dec 05 2024

Edited by Max Alekseyev, Jul 05 2010

Better name by Joerg Arndt, Aug 17 2013

A123513 Triangle read by rows: T(n,k) is the number of permutations of [n] having k small descents (n >= 1; 0 <= k <= n-1). A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) = 1.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 11, 9, 3, 1, 53, 44, 18, 4, 1, 309, 265, 110, 30, 5, 1, 2119, 1854, 795, 220, 45, 6, 1, 16687, 14833, 6489, 1855, 385, 63, 7, 1, 148329, 133496, 59332, 17304, 3710, 616, 84, 8, 1, 1468457, 1334961, 600732, 177996, 38934, 6678, 924, 108, 9, 1

Offset: 1

Emeric Deutsch, Oct 02 2006

This triangle is essentially A010027 (ascending pairs in permutations of [n]) with a different offset. The same triangle gives the number of permutations of [n] having k unit ascents (n >= 1; 0 <= k <= n-1). For permutations sorted by number of non-unitary (i.e., > 1) descents (also called "big" descents), see A120434. For permutations sorted by number of unitary moves (i.e., ascent or descent), see A001100. - Olivier Gérard, Oct 09 2007

With offsets n=0 (k=0) this is a binomial convolution triangle, a Sheffer triangle of the Appell type: ((exp(-x))/(1-x)^2),x). See the e.g.f. given below.

			Triangle starts:
     1;
     1,    1;
     3,    2,   1;
    11,    9,   3,   1;
    53,   44,  18,   4,  1;
   309,  265, 110,  30,  5, 1;
  2119, 1854, 795, 220, 45, 6, 1;
  ...
T(4,2)=3 because we have 14/3/2, 2/14/3 and 3/2/14 (the unit descents are shown by a /).
T(4,2)=3 because we have 14/3/2, 2/14/3 and 3/2/14 (the small descents are shown by a /).

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 179, Table 5.4 for S_{n,k} (without row n=1 and column k=0).
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263 (Table 7.5.1).

Alois P. Heinz, Rows n = 1..150, flattened
Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016. See Table 0.
FindStat - Combinatorial Statistic Finder, The number of adjacencies of a permutation
Sergey Kitaev and Philip B. Zhang, Distributions of mesh patterns of short lengths, arXiv:1811.07679 [math.CO], 2018.
J. Liese and J. Remmel, Q-analogues of the number of permutations with k-excedances, PU. M. A. Vol. 21 (2010), No. 2, pp. 285-320 (see E_{n,1}(x) in Table 1 p. 291).
F. Poussin, Énumération des permutations par nombre de marches, RAIRO, Informatique théorique, 13 no. 3, 1979, p. 251-255.

Cf. A000166, A000255, A000274, A000313, A001260.

Cf. A010027 (mirror image), A120434, A001100.

G:=exp(-x+t*x)/(1-x)^2: Gser:=simplify(series(G,x=0,15)): for n from 0 to 10 do P[n+1]:=sort(n!*coeff(Gser,x,n)) od: for n from 1 to 11 do seq(coeff(P[n],t,k),k=0..n-1) od; # yields sequence in triangular form

Needs["Combinatorica`"];
Table[Map[Count[#,1]&,Map[Differences,Permutations[n]]]//Distribution,{n,1,10}]//Grid
(* Geoffrey Critzer, Dec 15 2012 *)
T[n_, k_] := (n-1)! SeriesCoefficient[Exp[-x + t x]/(1-x)^2, {x, 0, n-1}, {t, 0, k}];
Table[T[n, k], {n, 1, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Sep 25 2019 *)
T[1,1]:=1;T[0,1]:=0;T[n_,1]:=T[n,1]=(n-1)T[n-1,1]+(n-2)T[n-2,1];T[n_,k_]:=T[n,k]=T[n-1, k-1](n-1)/(k-1);Flatten@Table[T[n,k],{n,1,10},{k,1,n}] (* Oliver Seipel, Dec 01 2024 *)

T(n,1) = A000255(n-1).
T(n,2) = A000166(n-1) (the derangement numbers).
T(n,3) = A000274(n).
T(n,4) = A000313(n).
T(n,5) = A001260(n);
G.f.: exp(-x+tx)/(1-x)^2 (if offset is 0), i.e., T(n,k)=(n-1)!*[x^(n-1) t^k]exp(-x+tx)/(1-x)^2.
T(n,k) = binomial(n-1,k)*A000255(n-1), n-1 >= k >= 0, else 0.

A000274 Number of permutations of length n with 2 consecutive ascending pairs.

Original entry on oeis.org

0, 0, 1, 3, 18, 110, 795, 6489, 59332, 600732, 6674805, 80765135, 1057289046, 14890154058, 224497707343, 3607998868005, 61576514013960, 1112225784377144, 21197714949305577, 425131949816628507, 8950146311929021210, 197350726178034917670, 4548464355722328578691

Offset: 1

N. J. A. Sloane, Simon Plouffe

From Emeric Deutsch, May 25 2009: (Start)
a(n) = number of excedances in all derangements of [n-1]. Example: a(5)=18 because the derangements of {1,2,3,4} are 4*123, 3*14*2, 3*4*12, 4*3*12, 2*14*3, 2*4*13, 2*3*4*1, 3*4*21, 4*3*21 with the 18 excedances marked. An excedance of a permutation p is a position i such that p(i)>i.
a(n) = Sum(k*A046739(n,k), k>=1).
(End)
Appears to be the inverse binomial transform of A001286 (filling the two leading zeros in there), then shifting one place to the right. - R. J. Mathar, Apr 04 2012

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 210 (divided by 2).
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 = 1..150
R. Mantaci and F. Rakotondrajao, Exceedingly deranging!, Advances in Appl. Math., 30 (2003), 177-188. [_Emeric Deutsch_, May 25 2009]

Cf. A010027, A000255, A000166, A000313, A001260, A001261.
A diagonal in triangle A010027.
Cf. A046739. [Emeric Deutsch, May 25 2009]
Cf. A145887, A145886.

a:= n->sum((n-1)!*sum((-1)^k/k!/2, j=1..n-1), k=0..n-1): seq(a(n), n=1..23); # Zerinvary Lajos, May 17 2007

Table[Subfactorial[n]*n/2, {n, 2, 20}] (* Zerinvary Lajos, Jul 09 2009 *)

a(n) = (1 + n) a(n - 1) + (3 + n) a(n - 2) + (3 - n) a(n - 3) + (2 - n) a(n - 4).
E.g.f.: x^2/2*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
a(n) = (n-1)^2/(n-2)*a(n-1)-(-1)^n*(n-1)/2, n>2, a(2)=0. - Vladeta Jovovic, Aug 31 2003
a(n) = (1/2){[n!/e] - [(n-1)!/e]} (conjectured).
a(n) = (n-1)*GAMMA(n,-1)*exp(-1)/2 where GAMMA = incomplete Gamma function. [Mark van Hoeij, Nov 11 2009]
a(n) = A145887(n-1) + A145886(n-1). - Anton Zakharov,  Aug 28 2016

Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

A000313 Number of permutations of length n with 3 consecutive ascending pairs.

Original entry on oeis.org

0, 0, 0, 1, 4, 30, 220, 1855, 17304, 177996, 2002440, 24474285, 323060540, 4581585866, 69487385604, 1122488536715, 19242660629360, 348933579412440, 6673354706262864, 134252194678935321, 2834212998777523380, 62651024183503148470, 1447238658638922729580

Offset: 1

N. J. A. Sloane

Temporary remark: there may be some issues with respect to the offset of this sequence in the formula and program sections. - Joerg Arndt, Nov 16 2014

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
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).

Todd Silvestri, Table of n, a(n) for n = 1..450 (first 100 terms from T. D. Noe)

Cf. A010027, A000255, A000166, A000274, A001260, A001261.

A diagonal in triangle A010027.

series(hypergeom([2,4],[],x/(x+1))/(x+1)^4, x=0, 30); # Mark van Hoeij, Nov 07 2011
a := n -> simplify(hypergeom([4-n,2],[],1))*(-1)^n*(n-1)*(n-2)*(n-3)/6: seq(a(n), n=1..23); # Peter Luschny, Nov 19 2014

Table[(n*(n + 1)!/6)*Sum[(-1)^k/k!, {k, 0, n}], {n, -1, 25}] (* T. D. Noe, Jun 19 2012 *)
a[1]:=0; a[n_Integer/;n>=2]:=(n-2) (n-1) Subfactorial[n-2]/6 (* Todd Silvestri, Nov 15 2014 *)

a = lambda n: (n-2)*(n-1)*sloane.A000166(n-2)/6 if n>2 else 0
[a(n) for n in range(1,24)] # Peter Luschny, Nov 19 2014

a(n) = (n*(n+1)!/6)*sum((-1)^k/k!, k=0..n).
a(n) = A065087(n+2)/3. - Zerinvary Lajos, May 25 2007
E.g.f.: x^3/3!*exp(-x)/(1-x)^2. - Vladeta Jovovic, Jan 03 2003
a(n) = round( (exp(-1)*(n+1)!+(-1)^n)*n/6 ). - Mark van Hoeij, Oct 25 2011
G.f.: hypergeom([2, 4],[],x/(x+1))/(x+1)^4. - Mark van Hoeij, Nov 07 2011
a(1) = 0, a(n) = (n-2)*(n-1)*(!(n-2))/6 = (n-2)*(n-1)*A000166(n-2)/6, for n >= 2. - Todd Silvestri, Nov 15 2014
a(n) = hypergeom([4-n,2],[],1)*(-1)^n*A000292(n-3). - Peter Luschny, Nov 19 2014
D-finite with recurrence (-n+4)*a(n) +(n-1)*(n-4)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 01 2022

More terms from Vladeta Jovovic, Jan 03 2003
Formula added by Sean A. Irvine, Nov 11 2010
Name clarified and offset changed by N. J. A. Sloane, Apr 12 2014

A059332 Determinant of n X n matrix A defined by A[i,j] = (i+j-1)! for 1 <= i,j <= n.

Original entry on oeis.org

1, 1, 2, 24, 3456, 9953280, 859963392000, 3120635156889600000, 634153008009974906880000000, 9278496603801318870491332608000000000, 12218100099725239100847669366019325952000000000000, 1769792823810713244721831122736499011207487815680000000000000000

Offset: 0

Noam Katz (noamkj(AT)hotmail.com), Jan 26 2001

nonn

Hankel transform of n! (A000142(n)) and of A003319. - Paul Barry, Oct 07 2008
Hankel transform of A000255. - Paul Barry, Apr 22 2009
Monotonic magmas of size n, i.e., magmas with elements labeled 1..n where product(i,j) >= max(i,j). - Chad Brewbaker, Nov 03 2013
Also called the bouncing factorial function. - Alexander Goebel, Apr 08 2020

			a(4) = 3456 because the relevant matrix is {1 2 6 24 / 2 6 24 120 / 6 24 120 720 / 24 120 720 5040 } and the determinant is 3456.

Alois P. Heinz, Table of n, a(n) for n = 0..32
Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5.
Googology Wiki, Bouncing Factorial

Cf. A010790, A385867.

Cf. A162014 and A055209. - Johannes W. Meijer, Jun 27 2009

with(linalg): Digits := 500: A059332 := proc(n) local A, i, j: A := array(1..n,1..n): for i from 1 to n do for j from 1 to n do A[i,j] := (i+j-1)! od: od: RETURN(det(A)) end: for n from 1 to 20 do printf(`%d,`, A059332(n)) od;
# second Maple program:
a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)*n!^2/n)
    end:
seq(a(n), n=0..12);  # Alois P. Heinz, Apr 29 2020

Table[n! BarnesG[n+1]^2, {n, 1, 10}] (* Jean-François Alcover, Sep 19 2016 *)

A059332(n)=matdet(matrix(n,n,i,j,(i+j-1)!)) \\ M. F. Hasler, Nov 03 2013

a(n) = 2^binomial(n,2)*prod(k=1,n-1, binomial(k+2,2)^(n-1-k)) \\ Ralf Stephan, Nov 04 2013

def mono_choices(a,b,n)
    n - [a,b].max
end
def all_mono_choices(n)
    accum =1
    0.upto(n-1) do |i|
        0.upto(n-1) do |j|
            accum = accum * mono_choices(i,j,n)
        end
    end
    accum
end
1.upto(12) do |k|
puts all_mono_choices(k)
end # Chad Brewbaker, Nov 03 2013

a(n) = a(n-1)*(n!)*(n-1)! for n >= 2 so a(n) = product k=1, 2, ..., n k!*(k-1)!.
a(n) = 2^C(n,2)*Product_{k=1..(n-1), C(k+2,2)^(n-1-k)}. - Paul Barry, Jan 15 2009
a(n) = n!*product(k!, k=0..n-1)^2. - Johannes W. Meijer, Jun 27 2009
a(n) ~ (2*Pi)^(n+1/2) * exp(1/6 - n - 3*n^2/2) * n^(n^2 + n + 1/3) / A^2, where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 01 2015

More terms from James Sellers, Jan 29 2001
Offset corrected. Comment and formula aligned with new offset by Johannes W. Meijer, Jun 24 2009
a(0)=1 prepended by Alois P. Heinz, Apr 08 2020

A096654 Denominators of self-convergents to 1/(e-2).

Original entry on oeis.org

1, 2, 8, 38, 222, 1522, 11986, 106542, 1054766, 11506538, 137119578, 1772006854, 24681524038, 368577425634, 5874202721042, 99515904921182, 1785757627196766, 33835407673201882, 675016383080377546, 14143200407398386678, 310507536216973671158, 7128173005328786885714

Offset: 0

Clark Kimberling, Jul 01 2004

The self-continued fraction of r>0 is here introduced as the sequence (b(0), b(1), b(2), ...) defined as follows: put r(0)=r, b(0)=[r(0)] and for n>=1, put r(n)=b(n-1)/(r(n-1)-b(n-1)) and b(n)=[r(n)]. This differs from simple continued fraction, for which r(n)=1/(r(n-1)-b(n-1)). Now r=lim(p(n)/q(n)), where p(0)=b(1), q(0)=1, p(1)=b(0)(b(1)+1), q(1)=b(1) and for n>=2, p(n)=b(n)*p(n-1)+b(n-1)*p(n-2), q(n)=b(n)*q(n-1)+b(n-1)*q(n-2); p(0),p(1),... are the numerators of the self-convergents to r; q(0),q(1),... are the denominators of the self-convergents to r. Thus A096654 is given by a(n)=(n+1)*a(n-1)+n*a(n-2), a(0)=1, a(1)=2.

Number of increasing runs of odd length in all permutations of [n+1]. Example: a(2) = 8 because we have (123), 13(2), (3)12, (2)13, 23(1), (3)(2)(1) (the runs of odd length are shown between parentheses). - Emeric Deutsch, Aug 29 2004

			a(2)=q(2)=3*2+2*1=8, a(3)=q(3)=4*8+3*2=38. The convergents p(0)/q(0) to p(4)/q(4) are 1/1, 3/2, 11/8, 53/38, 309/222.

Cf. A000255, A096655, A096656, A096657, A096658, A097591, A233590.

G:=(3-x-2*(1+x)*exp(-x))/(1-x)^3: Gser:=series(G,x=0,22): 1,seq(n!*coeff(Gser,x^n),n=1..21);

With[{g = (3 - x - 2*(1 + x)*Exp[-x])/(1 - x)^3},CoefficientList[Series[g, {x, 0, 21}], x]*Table[k!, {k, 0, 21}]] (* Shenghui Yang, Oct 15 2024 *)

x='x+O('x^66); Vec(serlaplace((3-x-2*(1+x)*exp(-x))/(1-x)^3)) /* Joerg Arndt, Aug 06 2012 */

prpr = 1
prev = 2
for n in range(2, 77):
    print(prpr, end=', ')
    curr = (n+1)*prev + n*prpr
    prpr = prev
    prev = curr
# Alex Ratushnyak, Aug 05 2012

a(n) = (n+1)*a(n-1) + n*a(n-2), with a(0)=1, a(1)=2. - Alex Ratushnyak, Aug 05 2012
E.g.f.: (3-x-2*(1+x)*exp(-x))/(1-x)^3. - Emeric Deutsch, Aug 29 2004
From Gary Detlefs, Apr 12 2010: (Start)
a(n) = A055596(n+1) + A055596(n+2).
a(n) = (n+1)!+(n+2)! -2*( A000166(n+1) + A000166(n+2)).
a(n) = (n+1)! - 2*floor(((n+1)!+1)/e) + (n+2)!-2*floor(((n+2)!+1)/e). (End)
a(n) = ((n+3)!-2*floor(((n+3)!+1)/e))/(n+2). - Gary Detlefs, Jul 11 2010 [corrected by Gary Detlefs, Oct 26 2020]
a(n) = Sum_{k=1..n+1} A097591(n+1,k). - Alois P. Heinz, Jul 03 2019

More terms from Emeric Deutsch, Aug 29 2004

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

A086892 Greatest common divisor of 2^n-1 and 3^n-1.

Original entry on oeis.org

1, 1, 1, 5, 1, 7, 1, 5, 1, 11, 23, 455, 1, 1, 1, 85, 1, 133, 1, 275, 1, 23, 47, 455, 1, 1, 1, 145, 1, 2387, 1, 85, 23, 1, 71, 23350145, 1, 1, 1, 11275, 1, 2107, 431, 115, 1, 47, 1, 750295, 1, 11, 1, 265, 1, 133, 23, 145, 1, 59, 1, 47322275, 1, 1, 1, 85, 1, 10787, 1, 5, 47, 781, 1

Offset: 1

Joseph H. Silverman (jhs(AT)math.brown.edu), Sep 18 2003

a(n) is a simple (the simplest?) example of a divisibility sequence associated to a rational point on an algebraic group of dimension larger than two. Specifically, it is the divisibility sequence associated to the point (2,3) on the two-dimensional torus G_m^2. Ailon and Rudnick conjecture that a(n) = 1 for infinitely many n.

According to Corvaja, a(n) < 2^n - 1 for all but finitely many n.

Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1. Math. Z. 243 (2003), no. 1, 79-84

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
N. Ailon, Z. Rudnick, Torsion points on curves and common divisors of a^k-1 and b^k-1, Acta Arith. 113 (2004), no. 1, 31-38.
Y. Bugeaud, P. Corvaja, U. Zannier, An upper bound for the G.C.D. of a^n-1 and b^n-1, Math. Z. 243 (2003), no. 1, 79-84
P. Corvaja, Greatest Common Divisors in Vojta's Conjecture: Arithmetic and Geometry, Journées Arithmétiques 2011.
Index to divisibility sequences

Cf. A000225, A000255, A003462, A024023, A260119.

a086892 n = a086892_list !! (n-1)
a086892_list = tail $ zipWith gcd a000225_list a003462_list
-- Reinhard Zumkeller, Jul 18 2015

[Gcd(2^n-1, 3^n-1): n in [1..75]]; // Vincenzo Librandi, Sep 02 2015

seq(igcd(2^n-1,3^n-1), n=1..100); # Robert Israel, Sep 02 2015

Table[GCD[2^n - 1, 3^n - 1], {n, 100}] (* Vincenzo Librandi, Sep 02 2015 *)

PARI
```
vector(100,n,gcd(2^n-1,3^n-1))
```

a(n) = gcd(2^n - 1, 3^n - 1).

a(n) = GCD(A000255(n), A003462(n)) = GCD(A000255(n), A024023(n)). - Reinhard Zumkeller, Mar 26 2004

Replaced arXiv URL with non-cached version by R. J. Mathar, Oct 23 2009

cp's OEIS Frontend

A090012 Permanent of (0,1)-matrix of size n X (n+d) with d=2 and n-1 zeros not on a line.

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A090016 Permanent of (0,1)-matrix of size n X (n+d) with d=6 and n-1 zeros not on a line.

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A153229 a(0) = 0, a(1) = 1, and for n >= 2, a(n) = (n-1) * a(n-2) + (n-2) * a(n-1).

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A123513 Triangle read by rows: T(n,k) is the number of permutations of [n] having k small descents (n >= 1; 0 <= k <= n-1). A small descent in a permutation (x_1,x_2,...,x_n) is a position i such that x_i - x_(i+1) = 1.

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A000274 Number of permutations of length n with 2 consecutive ascending pairs.

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A000313 Number of permutations of length n with 3 consecutive ascending pairs.

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