Philippe Flajolet - cp's OEIS frontend

User: Philippe Flajolet

Philippe Flajolet has authored 6 sequences.

A122949 Number of ordered pairs of permutations generating a transitive group.

1, 3, 26, 426, 11064, 413640, 20946960, 1377648720, 114078384000, 11611761920640, 1425189271161600, 207609729886944000, 35419018603306060800, 6996657393055480550400, 1584616114318716544665600, 407930516160959891683584000, 118458533875304716189544448000

Offset: 1

Philippe Flajolet, Oct 25 2006

nonn

From Dixon: The sequence is asymptotic to (n!)^2; when divided by n!^2, it has a high-order asymptotic contact with the probability that two randomly chosen permutations generate the symmetric group. Also: a(n)=(n-1)!*A003319(n+1), where A003319 is the number of connected [or indecomposable] permutations. The coefficients in the asymptotic expansion of a(n)/(n!)^2 are A113869 and in absolute value, they constitute A084357 (number of sets of sets of lists).

			a(2)=3 because there are 2!*2!=4 pairs of permutations, of which only [(1,1),(1,1)] does not generate a transitive group.

Seiichi Manyama, Table of n, a(n) for n = 1..253
John D. Dixon, Asymptotics of Generating the Symmetric and Alternating Groups, Electronic Journal of Combinatorics, vol 11(2), R56.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 139.

Cf. A003319, A084357, A113869.

series(log(add(n!*z^n,n=0..Order+2)),z=0):seq(coeff(%,z,j)*j!,j=0..Order);

max = 15; Drop[ CoefficientList[ Series[ Log[1 + Sum[n!*z^n, {n, 1, max}]], {z, 0, max}], z]* Range[0, max]!, 1](* Jean-François Alcover, Oct 05 2011 *)

N=20; x='x+O('x^N); Vec(serlaplace(log(sum(k=0, N, k!*x^k)))) \\ Seiichi Manyama, Mar 01 2019

Exponential generating function is: log(1+Sum_{n>=1}n!*z^n).

a(n) = (n!)^2 - (n-1)! * Sum_{k=1..n-1} a(k) * (n-k)! / (k-1)!. - Ilya Gutkovskiy, Jul 10 2020

More terms from Seiichi Manyama, Mar 01 2019

A102866 Number of finite languages over a binary alphabet (set of nonempty binary words of total length n).

Original entry on oeis.org

1, 2, 5, 16, 42, 116, 310, 816, 2121, 5466, 13937, 35248, 88494, 220644, 546778, 1347344, 3302780, 8057344, 19568892, 47329264, 114025786, 273709732, 654765342, 1561257968, 3711373005, 8797021714, 20794198581, 49024480880, 115292809910, 270495295636

Offset: 0

Philippe Flajolet, Mar 01 2005

nonn

Analogous to A034899 (which also enumerates multisets of words)

			a(2) = 5 because the sets are {a,b}, {aa}, {ab}, {ba}, {bb}.
a(3) = 16 because the sets are {a,aa}, {a,ab}, {a,ba}, {a,bb}, {b,aa}, {b,ab}, {b,ba}, {b,bb}, {aaa}, {aab}, {aba}, {abb}, {baa}, {bab}, {bba}, {bbb}.

Alois P. Heinz, Table of n, a(n) for n = 0..1000
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 64
Stefan Gerhold, Counting finite languages by total word length, INTEGERS 11 (2011), #A44.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 27.

Cf. A000079, A034899, A256142.
Column k=2 of A292804.
Row sums of A208741 and of A360634.

series(exp(add((-1)^(j-1)/j*(2*z^j)/(1-2*z^j),j=1..40)),z,40);

nn = 20; p = Product[(1 + x^i)^(2^i), {i, 1, nn}]; CoefficientList[Series[p, {x, 0, nn}], x] (* Geoffrey Critzer, Mar 07 2012 *)
CoefficientList[Series[E^Sum[(-1)^(k-1)/k*(2*x^k)/(1-2*x^k), {k,1,30}], {x, 0, 30}], x] (* Vaclav Kotesovec, Sep 13 2014 *)

G.f.: exp(Sum((-1)^(j-1)/j*(2*z^j)/(1-2*z^j), j=1..infinity)).
Asymptotics (Gerhold, 2011): a(n) ~ c * 2^(n-1)*exp(2*sqrt(n)-1/2) / (sqrt(Pi) * n^(3/4)), where c = exp( Sum_{k>=2} (-1)^(k-1)/(k*(2^(k-1)-1)) ) = 0.6602994483152065685... . - Vaclav Kotesovec, Sep 13 2014
Weigh transform of A000079. - Alois P. Heinz, Jun 25 2018

A077608 Number of compositions of n into twin primes (i.e., primes that are members of a twin prime pair, like 3, 5, 7, 11, 13, but not 2 or 23).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 75, 103, 125, 167, 220, 275, 370, 474, 610, 806, 1028, 1347, 1752, 2253, 2954, 3812, 4944, 6451, 8329, 10841, 14077, 18226, 23720, 30745, 39903, 51857, 67214, 87313, 113340, 147017, 190974

Offset: 0

Philippe Flajolet, Nov 11 2002

nonn

			a(15) = 8 since 15 = 11+7 = 7+11 = 5+13 = 13+5 = 3+5+7 = 3+7+5 = 5+3+7 = 5+7+3 = 7+3+5 = 7+5+3 and 3,5,7,11 belong to twin pairs.

Iain Fox, Table of n, a(n) for n = 0..8834 (first 1001 terms from Alois P. Heinz)
P. Flajolet, Publications

Cf. A001097, A002124, A023360.

A077608 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j)* subs([true=1,false=0],evalb(isprime(ithprime(j)-2) or isprime(ithprime(j)+2))),j=2..n+2)),z=0,n+1),z,n): end;

a[n_] := Coefficient[Series[ 1/(1 - Sum[z^Prime[j]*Boole[ PrimeQ[Prime[j] - 2] || PrimeQ[ Prime[j] + 2]], {j, 2, n + 2}]), {z, 0, n + 1}], z, n]; Table[a[n], {n, 0, 53}] (* Jean-François Alcover, Nov 09 2012, after Maple *)

ok(n)={isprime(n) && (isprime(n-2) || isprime(n+2))}
{my(n=60); Vec(1/(1-sum(k=1, n, if(ok(k), x^k, 0))) + O(x*x^n))} \\ Andrew Howroyd, Dec 28 2017

G.f.: 1/(1 - Sum_{k>=1} x^A001097(k)). - Andrew Howroyd, Dec 28 2017

A060941 Duchon's numbers: the number of paths of length 5n from the origin to the line y = 2x/3 with unit East and North steps that stay below the line or touch it.

Original entry on oeis.org

1, 2, 23, 377, 7229, 151491, 3361598, 77635093, 1846620581, 44930294909, 1113015378438, 27976770344941, 711771461238122, 18293652115906958, 474274581883631615, 12388371266483017545, 325714829431573496525, 8613086428709348334675, 228925936056388155632081

Offset: 0

Philippe Flajolet, May 12 2001

A generalization of the ballot numbers.

Alois P. Heinz, Table of n, a(n) for n = 0..500
C. Banderier, Home page
Cyril Banderier and Philippe Flajolet, Basic Analytic Combinatorics of Lattice Paths, Theoret. Comput. Sci. 281 (2002), 37-80.
D. Bevan, D. Levin, P. Nugent, J. Pantone, and L. Pudwell, Pattern avoidance in forests of binary shrubs, arXiv preprint arXiv:1510:08036 [math.CO], 2015.
Daniel Birmajer, Juan B. Gil, Peter R. W. McNamara, and Michael D. Weiner, Enumeration of colored Dyck paths via partial Bell polynomials, arXiv:1602.03550 [math.CO], 2016.
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62.
M. T. L. Bizley, Derivation of a new formula for the number of minimal lattice paths from (0, 0) to (km, kn) having just t contacts with the line my = nx and having no points above this line; and a proof of Grossman's formula for the number of paths which may touch but do not rise above this line, Journal of the Institute of Actuaries, Vol. 80, No. 1 (1954): 55-62. [Cached copy]
M. T. L. Bizley, Annotated copy of page 59
M. Bousquet-Mélou and A. Jehanne, Polynomial equations with one catalytic variable, algebraic series and map enumeration, arXiv:math/0504018 [math.CO], 2005.
P. Duchon, Home Page
Philippe Duchon, On the enumeration and generation of generalized Dyck words, Discrete Mathematics 225, 2000, 121-135.
Bryan Ek, Lattice Walk Enumeration, arXiv:1803.10920 [math.CO], 2018.
Bryan Ek, Unimodal Polynomials and Lattice Walk Enumeration with Experimental Mathematics, arXiv:1804.05933 [math.CO], 2018.
P. Flajolet, Home page
Don Knuth, 20th Anniversary Christmas Tree Lecture [A060941 is mentioned after about 65 minutes - _N. J. A. Sloane_, Dec 09 2014]
Michael Wallner, Combinatorics of lattice paths and tree-like structures (Dissertation, Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien), 2016.

Cf. A000108, A001450, A001764, A002294, A300386 - A300389, A322634.

See A293946 for a closely related sequence, also from the Bizley paper.

[&+[1/(5*n+i+1)*Binomial(5*n+1, n-i)*Binomial(5*n+2*i, i): i in [0..n]]: n in [0..30]]; // Vincenzo Librandi, Feb 12 2016

A060941 := n -> hypergeom([-n,5*n/2+1/2,5*n/2+1],[4*n+2,5*n+2],-4)* binomial(5*n,n)/(4*n+1); seq(simplify(A060941(n)),n=0..18); # Peter Luschny, Oct 05 2014

a[n_] := ((5n)!*(5n + 1)!*HypergeometricPFQRegularized[{-n, 5n/2 + 1/2, 5n/2 + 1}, {4n + 2, 5n + 2}, -4])/n!; a /@ Range[0, 16]
(* Jean-François Alcover, Jun 30 2011, after given formula *)

A060941 = lambda n : hypergeometric([-n,5*n/2+1/2,5*n/2+1],[4*n+2,5*n+2],-4)*gamma(1+5*n)/(gamma(1+n)*gamma(2+4*n))
[A060941(n).simplify() for n in range(19)] # Peter Luschny, Oct 05 2014

a(n) = Sum_{i=0..n} 1/(5*n+i+1) * C(5*n+1, n-i) * C(5*n+2*i, i).
a(n) = Sum_{i=0..2*n} (-1)^i/(5*i+1) * C((5*i+1)/2, i) * 1/(1+5*(2*n-i)) * C((1+5*(2*n-i))/2, 2*n-i).
G.f. A(z) satisfies: A(z) = 1+2*z*A^5-z*A^6+z*A^7+z^2*A^10. [Corrected by Bryan T. Ek, Oct 30 2017]
G.f.: A(z) = exp(C(5,2)*z/5 + C(10,4)*z^2/10 + C(15,6)*z^3/15 + ...). - Don Knuth, Oct 05 2014
Recurrence: 216*(n-1)*n*(2*n-1)*(3*n-4)*(3*n-2)*(3*n-1)*(3*n+1)*(6*n-1)*(6*n+1)*(5625*n^4 - 38550*n^3 + 97425*n^2 - 107784*n + 44044)*a(n) = 540*(n-1)*(3*n-4)*(3*n-2)*(126562500*n^10 - 1373625000*n^9 + 6557484375*n^8 - 18192221250*n^7 + 32549973750*n^6 - 39248008800*n^5 + 32203028675*n^4 - 17641491134*n^3 + 6113558828*n^2 - 1191132600*n + 96112128)*a(n-1) - 450*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(63281250*n^9 - 718453125*n^8 + 3556125000*n^7 - 10046426250*n^6 + 17765816250*n^5 - 20240090325*n^4 + 14698993900*n^3 - 6468702396*n^2 + 1533535184*n - 142988160)*a(n-2) + 78125*(n-2)*(5*n-14)*(5*n-13)*(5*n-12)*(5*n-11)*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(5625*n^4 - 16050*n^3 + 15525*n^2 - 6084*n + 760)*a(n-3). - Vaclav Kotesovec, Oct 05 2014
Asymptotics (Duchon, 2000): a(n) ~ c * (3125/108)^n / n^(3/2), where c = 0.0876612192439026461763141944768209255550234422281635788... (constant corrected, in the reference "On the enumeration and generation of generalized Dyck words", p.132 is a wrong value 0.0887). - Vaclav Kotesovec, Oct 05 2014, c = sqrt(5*(10^(2/3) - 5^(1/3)/2^(2/3) - 2))/(18*sqrt(Pi)). - Vaclav Kotesovec, Sep 16 2021
a(n) = Gamma(n+4/5)*Gamma(n+3/5)*Gamma(n+2/5)*3125^n*hypergeom([-n, (5/2)*n+1, (5/2)*n+1/2], [5*n+2, 4*n+2], -4)*Gamma(n+1/5)/ (Pi^2*csc((2/5)*Pi)*csc((1/5)*Pi)*Gamma(4*n+2)). - Robert Israel, Oct 05 2014
a(n) = A002294(n)*hypergeom([-n,5*n/2+1/2,5*n/2+1],[4*n+2,5*n+2],-4). - Peter Luschny, Oct 05 2014
O.g.f. A(x) satisfies: A(x)^5 = 1/x*series reversion( x/((1+x)*C(x))^5 ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See A001450. - Peter Bala, Oct 05 2015
The sequence defined by b(n) := [x^n] A(x)^n begins [1, 2, 50, 1415, 42258, 1300727, 40820837, 1298493730, ...] and conjecturally satisfies the congruence b(p) == b(1) (mod p^3) for prime p >= 7 (checked up to p = 101). [Added 23 Oct 2024: More generally, let r  be an integer and s a positive integer and define a sequence u(n) by u(n) = [x^(s*n)] A(x)^(r*n). Then we conjecture that the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) hold for all primes p >= 7 and positive integers n and k.] - Peter Bala, Sep 12 2021
Inductively define a family of sequences {a(i,n) : n >= 0}, i >= 1, by setting a(1,n) = a(n) and, for i >= 2, a(i,n) = [x^n] ( exp(Sum_{k >= 1} a(i-1,k)*x^k/k) )^n. We conjecture that the sequences {a(i,n) : n >= 0}, i >= 2, also satisfy the supercongruences u(n*p^k) == u(n*p^(k-1)) (mod p^(3*k)) for primes p >= 7 and positive integers n and k. - Peter Bala, Oct 24 2024

A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges.

Original entry on oeis.org

1, 2, 7, 33, 181, 1083, 6854, 45111, 305629, 2117283, 14929212, 106790500, 773035602, 5652275723, 41683912721, 309691336359, 2315772552485, 17415395593371, 131632335068744, 999423449413828, 7618960581522348, 58295017292748756, 447517868947619432, 3445923223190363608

Offset: 1

Philippe Flajolet, Apr 20 2000

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. Banderier and D. Merlini, Lattice paths with an infinite set of jumps, FPSAC02, Melbourne, 2002.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Guillermo Esteban, Clemens Huemer, Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
Philippe Flajolet, Source
Philippe Flajolet, Enumeration of planar configurations in computational geometry
P. Flajolet and M. Noy, Analytic combinatorics of noncrossing configurations, Discrete Math. 204 (1999), 203-229.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486, 502
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 8, arXiv:1709.08416 [math.CO], 2017.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.

Row sums of A094021.

Cf. A006013, A000108.

ZZ:=[F,{F=Union(Epsilon,ZB),ZB=Prod(Z1,P),P=Sequence(B),B=Prod(P,Z1,P),Z1=Prod(Z,F)}, unlabeled]: seq(count(ZZ,size=n),n=1..20); # Zerinvary Lajos, Apr 22 2007

a[n_] := (3*n-3)!/((n-1)!*(2*n-1)!)*HypergeometricPFQ[{1-2*n, 1-n, -n}, {3/2 - 3*n/2, 2 - 3*n/2}, -1/4]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 05 2012, after formula *)

N=33; x='x+O('x^N); Vec(serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x)))) \\ Joerg Arndt, May 25 2016

a(n) = Sum_{j=1..n} binomial(n, j-1) * binomial(3*n-2*j-1, n-j) / (2*n - j).
G.f. A(x) satisfies 2*A(x)^2=x*(1-sqrt(1-4*A(x)))*(1-A(x)). - Vladimir Kruchinin, Nov 25 2011
From Peter Bala, Nov 07 2015: (Start)
O.g.f. A(x) = revert(x/((1 + x)*C(x))), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f for the Catalan numbers A000108.
Row sums of A094021. (End)
Conjecture: -2(37*n-80) *(n-1) *(2*n-1) *a(n) +2*(592*n^3-3056*n^2+5045*n-2665) *a(n-1) +2*(148*n^3-986*n^2+2021*n-1255) *a(n-2) -5*(n-5) *(n-2) *(37*n-43) *a(n-3)=0. - R. J. Mathar, Apr 30 2018
a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 8.2246915409778560686084627753... is the real root of the equation 5 - 8*d - 32*d^2 + 4*d^3 = 0 and c = 0.07465927842190452347018812862935237... is the positive real root of the equation -125 + 22376*c^2 + 8880*c^4 + 592*c^6 = 0. - Vaclav Kotesovec, Apr 30 2018

A054726 Number of graphs with n nodes on a circle without crossing edges.

Original entry on oeis.org

1, 1, 2, 8, 48, 352, 2880, 25216, 231168, 2190848, 21292032, 211044352, 2125246464, 21681954816, 223623069696, 2327818174464, 24424842461184, 258054752698368, 2742964283768832, 29312424612462592, 314739971287154688, 3393951437605044224, 36739207546043105280

Offset: 0

Philippe Flajolet, Apr 20 2000

Related to Schröder's second problem.
A001006 gives number of ways of drawing any number of nonintersecting chords between n points on a circle, while this sequence gives number of ways of drawing noncrossing chords between n points on a circle. The difference is that nonintersection chords have no point in common, while noncrossing chords may share an endpoint. - David W. Wilson, Jan 30 2003
For n>0, a(n) = number of lattice paths from (0,0) to (n-1,n-1) that consist of steps (i,j), i,j nonnegative integers not both 0 and that stay strictly below the line y=x except at their endpoints. For example, a(3)=8 counts the paths with following step sequences: {(2, 2)}, {(2, 1), (0, 1)}, {(2, 0), (0, 2)}, {(2, 0), (0, 1), (0, 1)}, {(1, 0), (1, 2)}, {(1, 0), (1, 1), (0, 1)}, {(1, 0), (1, 0), (0, 2)}, {(1, 0), (1, 0), (0, 1), (0, 1)}. If the word "strictly" is replaced by "weakly", the counting sequence becomes A059435. - David Callan, Jun 07 2006
The nodes on the circle are distinguished by their positions but are otherwise unlabeled. - Lee A. Newberg, Aug 09 2011
From Gus Wiseman, Jun 22 2019: (Start)
Conjecture: Also the number of simple graphs with vertices {1..n} not containing any pair of nesting edges. Two edges {a,b}, {c,d} where a < b and c < d are nesting if a < c and b > d or a > c and b < d. For example, the a(0) = 1 through a(3) = 8 non-nesting edge-sets are:
  {}  {}  {}    {}
          {12}  {12}
                {13}
                {23}
                {12,13}
                {12,23}
                {13,23}
                {12,13,23}
Cf. A001519, A117662, A326244, A326256, A326257, A326279.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
F. Cazals, Combinatorics of Non-Crossing Configurations, Studies in Automatic Combinatorics, Volume II (1997).
Michael Drmota, Anna de Mier, and Marc Noy, Extremal statistics on non-crossing configurations, Discrete Math. 327 (2014), 103--117. MR3192420. See Eq. (3). - _N. J. A. Sloane_, May 18 2014
Guillermo Esteban, Clemens Huemer, and Rodrigo I. Silveira, New production matrices for geometric graphs, arXiv:2003.00524 [math.CO], 2020.
P. Flajolet and M. Noy, Analytic Combinatorics of Noncrossing Configurations, Discr. Math. 204 (1999), 203-229.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 486.
Samuele Giraudo, Combalgebraic structures on decorated cliques, Formal Power Series and Algebraic Combinatorics, Séminaire Lotharingien de Combinatoire, 78B.15, 2017, p. 10; arXiv:1709.08416 [math.CO], 2017.
Marco Kuhlmann, Tabulation of Noncrossing Acyclic Digraphs, arXiv preprint arXiv:1504.04993 [cs.DS], 2015.
Gus Wiseman, The a(4) = 48 non-crossing graphs.
Gus Wiseman, The a(5) = 352 non-crossing graphs.
Gus Wiseman, The a(4) = 48 non-nesting simple graphs.
Anssi Yli-Jyrä and Carlos Gómez-Rodríguez, Generic Axiomatization of Families of Noncrossing Graphs in Dependency Parsing, arXiv:1706.03357 [cs.CL], 2017.
Sequences related to chord diagrams

Cf. A006013, A001003.
Sequences related to chords in a circle: A001006, A054726, A006533, A006561, A006600, A007569, A007678. See also entries for chord diagrams in Index file.
Cf. A000108 (non-crossing set partitions), A000124, A006125, A007297 (connected case), A194560, A306438, A324167, A324169 (covering case), A324173, A326210.

with(combstruct): br:= {EA = Union(Sequence(EA, card >= 2), Prod(V, Sequence(EA), Sequence(EA))), V=Union(Prod(Z, G)), G=Union(Epsilon, Prod(Z, G), Prod(V,V,Sequence(EA), Sequence(EA), Sequence(Union(Sequence(EA,card>=1), Prod(V,Sequence(EA),Sequence(EA)))))) }; ggSeq := [seq(count([G, br], size=i), i=0..20)];

Join[{a = 1, b = 1}, Table[c = (6*(2*n - 3)*b)/n - (4*(n - 3) a)/n; a = b; b = c, {n, 1, 40}]] (* Vladimir Joseph Stephan Orlovsky, Jul 11 2011 *)
nn=8;
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xGus Wiseman, Feb 19 2019 *)

z='z+O('z^66); Vec( 1+3/2*z-z^2-z/2*sqrt(1-12*z+4*z^2) ) \\ Joerg Arndt, Mar 01 2014

a(n) = 2^n*A001003(n-2) for n>2.
From Lee A. Newberg, Aug 09 2011: (Start)
G.f.: 1 + (3/2)*z - z^2 - (z/2)*sqrt(1 - 12*z + 4*z^2);
D-finite with recurrence: a(n) = ((12*n-30)*a(n-1) - (4*n-16)*a(n-2)) / (n-1) for n>1. (End)
a(n) ~ 2^(n - 7/4) * (1 + sqrt(2))^(2*n-3) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Oct 11 2012, simplified Dec 24 2017
a(n) = 2^(n-2) * (Legendre_P(n-1, 3) - Legendre_P(n-3, 3))/(2*n - 3) = 2^n * (Legendre_P(n-1, 3) - 3*Legendre_P(n-2, 3))/(4*n - 8), both for n >= 3. - Peter Bala, May 06 2024

Offset changed to 0 by Lee A. Newberg, Aug 03 2011

cp's OEIS Frontend

User: Philippe Flajolet

A122949 Number of ordered pairs of permutations generating a transitive group.

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A102866 Number of finite languages over a binary alphabet (set of nonempty binary words of total length n).

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A077608 Number of compositions of n into twin primes (i.e., primes that are members of a twin prime pair, like 3, 5, 7, 11, 13, but not 2 or 23).

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Maple

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A060941 Duchon's numbers: the number of paths of length 5*n from the origin to the line y = 2*x/3 with unit East and North steps that stay below the line or touch it.

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Magma

Maple

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Sage

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A054727 Number of forests of rooted trees with n nodes on a circle without crossing edges.

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Maple

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A054726 Number of graphs with n nodes on a circle without crossing edges.

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Maple

Mathematica

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A060941 Duchon's numbers: the number of paths of length 5n from the origin to the line y = 2x/3 with unit East and North steps that stay below the line or touch it.