A049774 -id:A049774 - cp's OEIS frontend

A231210 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

1, 1, 2, 5, 1, 14, 9, 1, 46, 59, 14, 1, 177, 358, 164, 20, 1, 790, 2235, 1589, 398, 27, 1, 4024, 14658, 15034, 5659, 909, 35, 1, 23056, 103270, 139465, 77148, 17875, 2021, 44, 1, 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1, 1027850, 6315499

Offset: 0

Alois P. Heinz, Nov 05 2013

			T(3,1) = 1: 123.
T(4,0) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.
T(4,1) = 9: 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3421, 4123.
T(4,2) = 1: 1234.
T(5,2) = 14: 12354, 12453, 12543, 13452, 13542, 14532, 21345, 23451, 23541, 24531, 31245, 34521, 41235, 51234.
T(5,3) = 1: 12345.
Triangle T(n,k) begins:
:  0 :      1;
:  1 :      1;
:  2 :      2;
:  3 :      5,      1;
:  4 :     14,      9,       1;
:  5 :     46,     59,      14,      1;
:  6 :    177,    358,     164,     20,      1;
:  7 :    790,   2235,    1589,    398,     27,     1;
:  8 :   4024,  14658,   15034,   5659,    909,    35,    1;
:  9 :  23056, 103270,  139465,  77148,  17875,  2021,   44,  1;
: 10 : 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1;

Alois P. Heinz, Rows n = 0..142, flattened
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns

Columns k=0-2 give: A231211, A231228, A228422.
Row sums give: A000142.
Cf. A049774, A177479.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u+j-1, o-j, [2, 2, 2][t])*`if`(t=2, x, 1), j=1..o)+
      add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..14);

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[ Sum[b[u+j-1, o-j, {2, 2, 2}[[t]]]*If[t == 2, x, 1], {j, 1, o}] + Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Alois P. Heinz *)

A065429 Number of permutations of {1..n} which contain 3 consecutive terms in increasing order.

Original entry on oeis.org

0, 0, 1, 7, 50, 371, 3023, 26962, 263503, 2806759, 32439638, 404794391, 5429249279, 77941628854, 1193095348531, 19406686847167, 334372746780002, 6085085617645595, 116659595136911903, 2350442395504338154, 49658877772798776295, 1097945956312067172727

Offset: 1

Joe Keane (jgk(AT)jgk.org), Nov 16 2001

nonn

			a(3)=1 since 1,2,3 is only solution.

Alois P. Heinz, Table of n, a(n) for n = 1..450

Equals n! - A049774(n).

A206806 Sum_{0A002620(j) is the j-th quarter-square.

Original entry on oeis.org

1, 4, 13, 30, 62, 112, 190, 300, 455, 660, 931, 1274, 1708, 2240, 2892, 3672, 4605, 5700, 6985, 8470, 10186, 12144, 14378, 16900, 19747, 22932, 26495, 30450, 34840, 39680, 45016, 50864, 57273, 64260, 71877, 80142, 89110, 98800, 109270, 120540, 132671, 145684

Offset: 2

Clark Kimberling, Feb 15 2012

Partial sums of A049774. For a guide to related sequences, see A206817.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Cf. A206817, A002620.

[(108-36*n-n^2+n^4+(70*n-266)*Ceiling((3-n)/2)-(42*n-234)*Ceiling((3-n)/2)^2+(8*n-88)*Ceiling((3-n)/2)^3+12*Ceiling((3-n)/2)^4-4*n*Floor(n/2)-(12*n-12)*Floor(n/2)^2-(8*n-24)*Floor(n/2)^3+12*Floor(n/2)^4)/12: n in [2..50]]; // Wesley Ivan Hurt, Jul 10 2014

A206806:=n->add(i*(n-i)*(i-ceil((i-1)/2)), i=1..n): seq(A206806(n), n=2..50); # Wesley Ivan Hurt, Jul 10 2014

s[k_] := Floor[k/2]*Ceiling[k/2]; t[1] = 0;
Table[s[k], {k, 1, 20}]    (* A002620 *)
p[n_] := Sum[s[k], {k, 1, n}];
c[n_] := n*s[n] - p[n];
t[n_] := t[n - 1] + (n - 1) s[n] - p[n - 1]
Table[c[n], {n, 2, 50}]    (* A049774 *)
f = Flatten[Table[t[n], {n, 2, 50}]]  (* A206806 *)
Table[Sum[i (n - i) (i - Ceiling[(i - 1)/2]), {i, n}], {n, 2, 50}] (* Wesley Ivan Hurt, Jul 10 2014 *)
CoefficientList[Series[-(2 x^2 + x + 1)/((x - 1)^5 (x + 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jul 10 2014 *)

vector(100, n, ((n+1)*(1+3*(-1)^(n+1)-2*(n+1)+2*(n+1)^2+2*(n+1)^3))/48) \\ Colin Barker, Jul 10 2014

Vec(-x^2*(2*x^2+x+1)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Jul 10 2014

[sum([sum([floor(k^2/4)-floor(j^2/4) for j in range(1,k)]) for k in range(2,n+1)]) for n in range(2,44)] # Danny Rorabaugh, Apr 18 2015

From Wesley Ivan Hurt, Jul 10 2014: (Start)
a(n) = Sum_{i=1..n} i * (n-i) * (i-ceiling((i-1)/2)).
a(n) = (108 - 36n - n^2 + n^4 + (70n - 266) * ceiling((3 - n)/2) - (42n - 234) * ceiling((3 - n)/2)^2 + (8n - 88) * ceiling((3 - n)/2)^3 + 12 * ceiling((3 - n)/2)^4 - 4n * floor(n/2) - (12n - 12) * floor(n/2)^2 - (8n - 24) * floor(n/2)^3 + 12 * floor(n/2)^4) / 12. (End)
a(n) = (n*(1+3*(-1)^n-2*n+2*n^2+2*n^3))/48. - Colin Barker, Jul 10 2014
G.f.: -x^2*(2*x^2+x+1) / ((x-1)^5*(x+1)^2). - Colin Barker, Jul 10 2014

A183611 E.g.f. satisfies: A'(x) = A(x)^2 + x*A(x)^3, with A(0) = 1.

Original entry on oeis.org

1, 1, 3, 14, 91, 756, 7657, 91504, 1260441, 19663280, 342669691, 6597811584, 139094618467, 3186675803584, 78834061767825, 2094418664339456, 59474007876381553, 1797637447068293376, 57623116235327599411

Offset: 0

Paul D. Hanna, Mar 21 2011

nonn

			E.g.f.: A(x) = 1 + x + 3*x^2/2! + 14*x^3/3! + 91*x^4/4! +...
A'(x) = 1 + 3*x + 14*x^2/2! + 91*x^3/3! + 756*x^4/4! +...
A(x)^2 = 1 + 2*x + 8*x^2/2! + 46*x^3/3! + 348*x^4/4! + 3262*x^5/5! +...
A(x)^3 = 1 + 3*x + 15*x^2/2! + 102*x^3/3! + 879*x^4/4! + 4395*x^5/5! +...
E.g.f. A(x) = d/dx Series_Reversion(G(x)) where G(x) begins:
G(x) = x - x^2/2! + x^4/4! - x^5/5! + x^7/7! - x^8/8! + x^10/10! - x^11/11! +...
The series reversion of G(x) begins:
x + x^2/2! + 3*x^3/3! + 14*x^4/4! + 91*x^5/5! + 756*x^6/6! +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..400
V. Dotsenko, Pattern avoidance in labelled trees, arXiv preprint arXiv:1110.0844 [math.CO], 2011-2012.

Cf. A199670, A049774.

terms = 20; A[_] = 0;
Do[A[x_] = 1+Integrate[A[x]^2 + x A[x]^3, x]+O[x]^terms // Normal, terms];
CoefficientList[A[x], x] Range[0, terms-1]! (* Jean-François Alcover, Oct 27 2018 *)

{a(n)=local(A=1);for(n=0,n,A=1+A*intformal(1+x*A+x*O(x^n)));n!*polcoeff(A,n)}

{a(n)=n!*polcoeff(deriv(serreverse(sum(m=1,n\3+1,x^(3*m-2)/(3*m-2)!-x^(3*m-1)/(3*m-1)!+x^2*O(x^n)))),n)}

E.g.f.: A(x) = 1 + A(x)*[Integral 1 + x*A(x) dx], where the integration does not include the constant term.
E.g.f.: d/dx Series_Reversion(Sum_{n>=1} x^(3*n-2)/(3*n-2)! - x^(3*n-1)/(3*n-1)!).
a(n) ~ n^n * exp(Pi*(n+1)/(3*sqrt(3))-n). - Vaclav Kotesovec, Feb 19 2014

A231211 Number of permutations of [n] avoiding simultaneously consecutive patterns 123, 1432, 2431, and 3421.

Original entry on oeis.org

1, 1, 2, 5, 14, 46, 177, 790, 4024, 23056, 146777, 1027850, 7852184, 64985116, 579191277, 5530869310, 56336971744, 609708912976, 6986749484177, 84510154473170, 1076016705993704, 14385283719409636, 201475033030143477, 2950048762311387430, 45073424916825354064

Offset: 0

Alois P. Heinz, Nov 05 2013

nonn

Number of permutations of [n] avoiding simultaneously consecutive step patterns up, up and up, down, down.

			a(3) = 5: 132, 213, 231, 312, 321.
a(4) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.
a(5) = 46: 13254, 14253, 14352, ..., 54231, 54312, 54321.
a(6) = 177: 132546, 132645, 142536, ..., 654231, 654312, 654321.

Alois P. Heinz, Table of n, a(n) for n = 0..250
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes
S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns

Column k=0 of A231210.

Cf. A049774, A177479.

b:= proc(u, o, t) option remember; `if`(t=4, 0, `if`(u+o=0, 1,
      add(b(u+j-1, o-j, [2, 4, 2][t]), j=1..o)+
      add(b(u-j, o+j-1, [1, 3, 4][t]), j=1..u)))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);
# second Maple program
n:=40: c[0,0]:=1: for i to n-1 do c[i,0]:=0 end do: for i to n-1 do for j to i do c[i,j] := c[i,j-1] + c[i-1,i-j] + 1 end do end do: 1, seq(c[k, k]/2, k=1..n-1); # Sergei N. Gladkovskii, Jul 27 2015

b[u_, o_, t_] := b[u, o, t] = If[t == 4, 0, If[u + o == 0, 1,
    Sum[b[u + j - 1, o - j, {2, 4, 2}[[t]]], {j, 1, o}] +
    Sum[b[u - j, o + j - 1, {1, 3, 4}[[t]]], {j, 1, u}]]];
a[n_] := b[n, 0, 1];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 22 2020, after Alois P. Heinz *)

a(n) ~ (1+exp(Pi/2)) * (2/Pi)^(n+1) * n!. - Vaclav Kotesovec, Aug 28 2014

A232864 Number of permutations of n elements not cyclically containing the consecutive pattern 123.

Original entry on oeis.org

1, 1, 2, 3, 12, 45, 234, 1323, 8856, 65529, 543510, 4937031, 49030596, 526930677, 6101871426, 75686176035, 1001517264432, 14079895613937, 209594037600558, 3293305758743679, 54470994630103260, 945988795762018029, 17211193919411902938, 327371367293394753627

Offset: 0

Richard Ehrenborg, Dec 01 2013

nonn

			a(4) = 12 comes from the 3 permutations 1324, 1423 and 1432, and by cyclically shifting we obtain 3 * 4 = 12 permutations.

Alois P. Heinz, Table of n, a(n) for n = 0..200
R. Ehrenborg, Cyclically consecutive permutation avoidance, arXiv:1312.2051 [math.CO], 2013.

Cf. A049774, A080635.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<2, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 1), j=1..u))
    end:
a:= n-> `if`(n=0, 1, n*b(0, n-1, 1)):
seq(a(n), n=0..25);  # Alois P. Heinz, Dec 01 2013

b[u_,o_,t_] := b[u, o, t] = If[u+o==0, 1, If[t<2, Sum[b[u+j-1, o-j, t+1], {j, 1, o}], 0] + Sum[b[u-j, o+j-1, 1], {j, 1, u}]];
a[n_]:= If[n==0, 1, n*b[0, n-1, 1]];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 14 2017, after Alois P. Heinz *)

a(n) = n! * Sum_{k=-oo..oo} (sqrt(3)/(2*Pi*(k+1/3)))^n for n >= 2.

a(n) = A080635(n-1)*n for n>0. - Alois P. Heinz, Dec 01 2013

A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001599, 6227020775, 87178290682, 1307674357710, 20922789683040, 355687423926240, 6402373618334400, 121645098513933120, 2432901965590252800, 51090941178938707200, 1124000703770606323200

Offset: 0

Vaclav Kotesovec, Jan 31 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A049774, A117158, A177523, A177533, A177553, A230051, A230231, A230232, A230233.

Column k=2047 of A242784.

b:= proc(u, o, t) option remember; `if`(u+o=0, 1,
      `if`(t<10, add(b(u+j-1, o-j, t+1), j=1..o), 0)+
      add(b(u-j, o+j-1, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30); # after Alois P. Heinz

CoefficientList[Series[6 / (Exp[-x] + Cos[x] + 2*Cos[x/2] * Cosh[Sqrt[3]*x/2] - Cosh[Sqrt[3]*x/2]*Sin[x/2] - Sin[x] + Cosh[x/2] * (2*Cos[Sqrt[3]*x/2] - Sqrt[3]*Sin[Sqrt[3]*x/2]) - Cos[Sqrt[3]*x/2]*Sinh[x/2] - Sqrt[3]*Cos[x/2]*Sinh[Sqrt[3]*x/2]), {x, 0, 25}], x] * Range[0, 25]!

E.g.f.: 1 / Sum_{n>=0} (12*n+1-x)*x^(12*n)/(12*n+1)!.
E.g.f.: 6 / (exp(-x) + cos(x) + 2*cos(x/2)*cosh(sqrt(3)*x/2) - cosh(sqrt(3)*x/2)*sin(x/2) - sin(x) + cosh(x/2)*(2*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)) - cos(sqrt(3)*x/2)*sinh(x/2) - sqrt(3)*cos(x/2)*sinh(sqrt(3)*x/2)).
a(n)/n! ~ c * (1/r)^n, where r = 1.0000000019270853046730165249753673978954992128247736041276... is the root of the equation Sum_{n>=0} (r^(12*n)/(12*n)! - r^(12*n+1)/(12*n+1)!) = 0, equivalently root of the equation exp(-r) + cos(r) + 2*cos(r/2)*cosh(sqrt(3)*r/2) - cosh(sqrt(3)*r/2)*sin(r/2) - sin(r) + cosh(r/2)*(2*cos(sqrt(3)*r/2) - sqrt(3)*sin(sqrt(3)*r/2)) - cos(sqrt(3)*r/2)*sinh(r/2) - sqrt(3)*cos(r/2)*sinh(sqrt(3)*r/2) = 0, c = 3/(r*sqrt((cosh(sqrt(3)*r/2) * sin(r/2) + sin(r))^2 + 2*sqrt(3)*cosh(r/2) * (cosh(sqrt(3)*r/2) * sin(r/2) + sin(r)) * sin(sqrt(3)*r/2) + 3*cosh(r/2)^2 * sin((sqrt(3)*r)/2)^2)) = 1.0000000210373483515818712802156496756788404534079689145773611990529818919... .

A328504 Number of inversion sequences of length n avoiding the consecutive pattern 010.

Original entry on oeis.org

1, 1, 2, 5, 17, 76, 417, 2701, 20199, 171329, 1624851, 17036586, 195685618, 2443572835, 32959210808, 477542545691, 7396931591165, 121976733648960, 2133460758692093, 39450254899737811, 768950119933799815, 15757352298761474101, 338663233082663363407

Offset: 0

Vaclav Kotesovec and Juan S. Auli, Oct 17 2019

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..250
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli, Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 4.

Cf. A049774, A052169, A071075, A200404.

b:= proc(n, j, t) option remember; `if`(n=0, 1, add(
      `if`(i>=j or i<>t, b(n-1, i, j), 0), i=1..n))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..25);  # Alois P. Heinz, Oct 18 2019

b[n_, j_, t_] := b[n, j, t] = If[n == 0, 1, Sum[If[i >= j || i != t, b[n - 1, i, j], 0], {i, 1, n}]];
a[n_] := b[n, 0, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Mar 12 2020, after Alois P. Heinz *)

a(n) ~ n! * c / sqrt(n), where c = 1.410641128930866501817126119... - Vaclav Kotesovec, Oct 19 2019

A182822 Exponential Riordan array, defining orthogonal polynomials related to permutations without double falls.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 5, 12, 6, 1, 17, 53, 39, 10, 1, 70, 279, 260, 95, 15, 1, 349, 1668, 1914, 880, 195, 21, 1, 2017, 11341, 15330, 8554, 2380, 357, 28, 1, 13358, 86019, 134317, 87626, 29379, 5530, 602, 36, 1, 99377, 722664, 1277604, 954885, 372771, 84231, 11508, 954, 45, 1, 822041, 6655121, 13149441, 11061480, 4924515, 1292445, 211533, 22020, 1440, 55, 1

Offset: 0

Paul Barry, Dec 05 2010

Inverse is the coefficient array for the orthogonal polynomials P(0,x) = 1, P(1,x) = x-1, P(n,x) = (x-n)*P(n-1,x) - (n-1)^2*P(n-2,x). Inverse is A182823. First column is A049774.

			Triangle begins
      1;
      1,     1;
      2,     3,      1;
      5,    12,      6,     1;
     17,    53,     39,    10,     1;
     70,   279,    260,    95,    15,    1;
    349,  1668,   1914,   880,   195,   21,   1;
   2017, 11341,  15330,  8554,  2380,  357,  28,  1;
  13358, 86019, 134317, 87626, 29379, 5530, 602, 36, 1;
Production matrix is
  1, 1;
  1, 2, 1;
  0, 4, 3,  1;
  0, 0, 9,  4,  1;
  0, 0, 0, 16,  5,  1;
  0, 0, 0,  0, 25,  6,  1;
  0, 0, 0,  0,  0, 36,  7,  1;
  0, 0, 0,  0,  0,  0, 49,  8,  1;
  0, 0, 0,  0,  0,  0,  0, 64,  9,   1;
  0, 0, 0,  0,  0,  0,  0,  0, 81,  10,  1;
  0, 0, 0,  0,  0,  0,  0,  0,  0, 100, 11, 1;

Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.

dim = 11; M[n_, n_] = 1; M[n_ /; 0 <= n <= dim-1, k_ /; 0 <= k <= dim-1] := M[n, k] = M[n-1, k-1] + (k+1)*M[n-1, k] + (k+1)^2*M[n-1, k+1]; M[, ] = 0;
Table[M[n, k], {n, 0, dim-1}, {k, 0, n}] (* Jean-François Alcover, Jun 18 2019 *)

def A182822_triangle(dim):
    T = matrix(ZZ,dim,dim)
    for n in (0..dim-1): T[n,n] = 1
    for n in (1..dim-1):
        for k in (0..n-1):
            T[n,k] = T[n-1,k-1]+(k+1)*T[n-1,k]+(k+1)^2*T[n-1,k+1]
    return T
A182822_triangle(9) # Peter Luschny, Sep 19 2012

Exponential Riordan array [exp(x/2)/(cos(sqrt(3)x/2)-sin(sqrt(3)x/2)/sqrt(3)), 2*sin(sqrt(3)x/2)/(sqrt(3)*cos(sqrt(3)x/2)-sin(sqrt(3)x/2))].
From Werner Schulte, Mar 27 2022: (Start)
T(n,k) = T(n-1,k-1) + (k+1) * T(n-1,k) + (k+1)^2 * T(n-1,k+1) for n > 0 with initial values T(0,0) = 1 and T(i,j) = 0 if j < 0 or i < j (see the Sage program below).
The row polynomials p(n,x) = Sum_{k=0..n} T(n,k) * x^k satisfy recurrence equation p(n,x) = (1+x) * (p(n-1,x) + p'(n-1,x)) + x * p"(n-1,x) for n > 0 with initial value p(0,x) = 1 where p' and p" are first and second derivative of p. (End)

A199675 Expansion of e.g.f. 1/(exp(-x) - Sum_{n>=0} (-x)^(3n+2)/(3n+2)!).

Original entry on oeis.org

1, 1, 2, 7, 31, 170, 1129, 8737, 77198, 767683, 8482519, 103093958, 1366897597, 19633740673, 303706037546, 5033465370031, 88983532209967, 1671402633292562, 33241154368669921, 697834148797749601, 15420722865332961206, 357805114894717632331, 8697446048869287663271

Offset: 0

Paul D. Hanna, Nov 09 2011

nonn

			E.g.f.: A(x) = 1 + x + 2*x^2/2! + 7*x^3/3! + 31*x^4/4! + 170*x^5/5! +...
where
A(x) = 1/(1 - x - x^3/3! + x^4/4! + x^6/6! - x^7/7! - x^9/9! + x^10/10! +...).

Cf. A049774, A199670.

{a(n)=n!*polcoeff(1/(exp(-x+x*O(x^n)) - sum(m=0, n\3, (-x)^(3*m+2)/(3*m+2)! )), n)}

{a(n)=n!*polcoeff(1/(sum(m=0, n\3+1, (-x)^(3*m)/(3*m)! + (-x)^(3*m+1)/(3*m+1)! +x^2*O(x^n))), n)}

E.g.f.: A(x) = 1/Q(0); Q(k) = 1-x/((3*k+1)-(x^2)*(3*k+1)/((x^2)+3*(3*k+2)*(k+1)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Nov 26 2011

cp's OEIS Frontend

A231210 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

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A065429 Number of permutations of {1..n} which contain 3 consecutive terms in increasing order.

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A206806 Sum_{0A002620(j) is the j-th quarter-square.

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A183611 E.g.f. satisfies: A'(x) = A(x)^2 + x*A(x)^3, with A(0) = 1.

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A231211 Number of permutations of [n] avoiding simultaneously consecutive patterns 123, 1432, 2431, and 3421.

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A232864 Number of permutations of n elements not cyclically containing the consecutive pattern 123.

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A254523 Number of permutations of [n] avoiding adjacent step pattern {up}^11.

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A328504 Number of inversion sequences of length n avoiding the consecutive pattern 010.

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A199675 Expansion of e.g.f. 1/(exp(-x) - Sum_{n>=0} (-x)^(3n+2)/(3n+2)!).