A111004 -id:A111004 - cp's OEIS frontend

A049774 Number of permutations of n elements not containing the consecutive pattern 123.

1, 1, 2, 5, 17, 70, 349, 2017, 13358, 99377, 822041, 7477162, 74207209, 797771521, 9236662346, 114579019469, 1516103040833, 21314681315998, 317288088082405, 4985505271920097, 82459612672301846, 1432064398910663705, 26054771465540507273, 495583804405888997218

Offset: 0

Tuwani A. Tshifhumulo (tat(AT)caddy.univen.ac.za)

Permutations on n letters without double falls. A permutation w has a double fall at k if w(k) > w(k+1) > w(k+2) and has an initial fall if w(1) > w(2).
Hankel transform is A055209. - Paul Barry, Jan 12 2009
Increasing colored 1-2 trees of order n with choice of two colors for the right branches of the vertices of outdegree 2 except those vertices on the path from the root to the leftmost leaf. - Wenjin Woan, May 21 2011

			Permutations without double increase and without pattern 123:
a(3) = 5: 132, 213, 231, 312, 321.
a(4) = 17: 1324, 1423, 1432, 2143, 2314, 2413, 2431, 3142, 3214, 3241, 3412, 3421, 4132, 4213, 4231, 4312, 4321.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pp. 156-157.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (5.2.17).

Alois P. Heinz, Table of n, a(n) for n = 0..464 (first 201 terms from Ray Chandler)
Martin Aigner, Catalan and other numbers: a recurrent theme, in Algebraic Combinatorics and Computer Science, a Tribute to Gian-Carlo Rota, pp.347-390, Springer, 2001.
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 1.
Paul Barry, Constructing Exponential Riordan Arrays from Their A and Z Sequences, Journal of Integer Sequences, 17 (2014), #14.2.6.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Nicolas Basset, Counting and generating permutations using timed languages, HAL Id: hal-00820373, 2013.
A. Baxter, B. Nakamura, and D. Zeilberger. Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes; Local copy [Pdf file only, no active links].
S. Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns arXiv:math/0505254 [math.CO], 2015.
S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-123.
Steven Finch, Pattern-Avoiding Permutations [Archived version]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]
Ira M. Gessel and Yan Zhuang, Counting permutations by alternating descents , arXiv:1408.1886 [math.CO], 2014. See Eq. (3). - _N. J. A. Sloane_, Aug 11 2014
Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14.
Mingjia Yang and Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Christopher Zhu, Enumerating Permutations and Rim Hooks Characterized by Double Descent Sets, arXiv:1910.12818 [math.CO], 2019.

Cf. A065429, A080635, A111004, A117158, A177523, A177533.
Column k=0 of A162975.
Column k=3 of A242784.
Equals 1 + A000303. - Greg Dresden, Feb 22 2020

b:= proc(u, o, t) option remember;
     `if`(u+o=0, 1, add(b(u-j, o+j-1, 0), j=1..u)+
     `if`(t=1, 0,   add(b(u+j-1, o-j, 1), j=1..o)))
    end:
a:= n-> b(n, 0$2):
seq(a(n), n=0..23);  # Alois P. Heinz, Nov 04 2021

Table[Simplify[ n! SeriesCoefficient[ Series[ Sqrt[3] Exp[x/2]/(Sqrt[3] Cos[Sqrt[3]/2 x] - Sin[Sqrt[3]/2 x]), {x, 0, n}], n] ], {n, 0, 40}]
(* Second program: *)
b[u_, o_, t_, k_] := b[u, o, t, k] = If[t == k, (u + o)!, If[Max[t, u] + o < k, 0, Sum[b[u + j - 1, o - j, t + 1, k], {j, 1, o}] + Sum[b[u - j, o + j - 1, 1, k], {j, 1, u}]]];
a[n_] := b[0, n, 0, 2] - b[0, n, 0, 3] + 1;
a /@ Range[0, 40] (* Jean-François Alcover, Nov 09 2020, after Alois P. Heinz in A000303 *)

E.g.f.: 1/Sum_{i>=0} (x^(3*i)/(3*i)! - x^(3*i+1)/(3*i+1)!). [Corrected g.f. --> e.g.f. by Vaclav Kotesovec, Feb 15 2015]
Equivalently, e.g.f.: exp(x/2) * r / sin(r*x + (2/3)*Pi) where r = sqrt(3)/2. This has simple poles at (3*m+1)*x0 where x0 = Pi/sqrt(6.75) = 1.2092 approximately and m is an arbitrary integer. This yields the asymptotic expansion a(n)/n! ~ x0^(-n-1) * Sum((-1)^m * E^(3*m+1) / (3*m+1)^(n+1)) where E = exp(x0/2) = 1.8305+ and m ranges over all integers. - Noam D. Elkies, Nov 15 2001
E.g.f.: sqrt(3)*exp(x/2)/(sqrt(3)*cos(x*sqrt(3)/2) - sin(x*sqrt(3)/2) ); a(n+1) = Sum_{k=0..n} binomial(n, k)*a(k)*b(n-k) where b(n) = number of n-permutations without double falls and without initial falls. - Emanuele Munarini, Feb 28 2003
O.g.f.: A(x) = 1/(1 - x - x^2/(1 - 2*x - 4*x^2/(1 - 3*x - 9*x^2/(1 - ... - n*x - n^2*x^2/(1 - ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
a(n) = leftmost column term of M^n*V, where M = an infinite tridiagonal matrix with (1,2,3,...) in the super, sub, and main diagonals and the rest zeros. V = the vector [1,0,0,0,...]. - Gary W. Adamson, Jun 16 2011
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 25 2011
a(n) ~ n! * exp(Pi/(3*sqrt(3))) * (3*sqrt(3)/(2*Pi))^(n+1). - Vaclav Kotesovec, Jul 28 2013
E.g.f.: T(0)/(1-x), where T(k) = 1 - x^2*(k+1)^2/( x^2*(k+1)^2 - (1-x-x*k)*(1-2*x-x*k)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 17 2013

Corrected and extended by Vladeta Jovovic, Apr 14 2001

A117158 Number of permutations avoiding the consecutive pattern 1234.

Original entry on oeis.org

1, 1, 2, 6, 23, 111, 642, 4326, 33333, 288901, 2782082, 29471046, 340568843, 4263603891, 57482264322, 830335952166, 12793889924553, 209449977967081, 3630626729775362, 66429958806679686, 1279448352687538463, 25874432578888440471, 548178875969847203202

Offset: 0

Steven Finch, Apr 26 2006

nonn

a(n) is the number of permutations on [n] that avoid the consecutive pattern 1234. It is the same as the number of permutations which avoid 4321.

F. N. David and D. E. Barton, Combinatorial Chance, Hafner, New York, 1962, pages 156-157.

Alois P. Heinz, Table of n, a(n) for n = 0..450 (terms n = 0..60 from Ray Chandler)
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes.
Sergi Elizalde, Asymptotic enumeration of permutations avoiding generalized patterns, Adv. Appl. Math. 36 (2006), 138-155.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Steven Finch, Pattern-Avoiding Permutations. [Archived version]
Steven Finch, Pattern-Avoiding Permutations. [Cached copy, with permission]
Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory A 53 (1990), 257-285.
Ira M. Gessel, Yan Zhuang, Counting permutations by alternating descents, arXiv:1408.1886 [math.CO], 2014. See displayed equation before Eq. (3) and set m=4. - _N. J. A. Sloane_, Aug 11 2014
Kaarel Hänni, Asymptotics of descent functions, arXiv:2011.14360 [math.CO], Nov 29 2020, p. 14.
Mingjia Yang, Doron Zeilberger, Increasing Consecutive Patterns in Words, arXiv:1805.06077 [math.CO], 2018.
Mingjia Yang, An experimental walk in patterns, partitions, and words, Ph. D. Dissertation, Rutgers University (2020).

Cf. A022558, A049774, A111004, A113228, A113229, A117156, A117226, A177523, A177533, A201692, A201693, A230051, A324132.
Column k=0 of A220183.
Column k=7 of A242784.

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, 0), j=1..u))
    end:
a:= n-> b(n, 0, 0):
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 07 2013

a[n_]:=Coefficient[Series[2/(Cos[x]-Sin[x]+Exp[ -x]),{x,0,30}],x^n]*n!
(* second program: *)
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, 0], {j, 1, u}]]; a[n_] := b[n, 0, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

From Sergei N. Gladkovskii, Nov 30 2011: (Start)
E.g.f.: 2/(exp(-x) + cos(x) - sin(x)) = 1/W(0) with continued fraction
W(k) = 1 + (x^(2*k))/(f + f*x/(4*k + 1 - x - (4*k + 1)*b/R)), where R := x^(2*k) + b -(x^(4*k+1))/(c + (x^(2*k+1)) + x*c/T); T := 4*k + 3 - x - (4*k + 3)*d/(d +(x^(2*k+1))/W(k+1)), and
f := (4*k)!/(2*k)!; b := (4*k + 1)!/(2*k + 1)!; c := (4*k + 2)!/(2*k + 1)!; and d :=(4*k + 3)!/(2*k + 2)!. (End)
a(n) ~ n! / (sin(r)*r^(n+1)), where r = 1.0384156372665563... is the root of the equation exp(-r) + cos(r) = sin(r). - Vaclav Kotesovec, Dec 11 2013

A240885 Decimal expansion of the unique positive solution of Integral_{0..x} exp(-t^2/2) dt = 1.

Original entry on oeis.org

1, 2, 7, 5, 5, 4, 7, 7, 3, 6, 4, 1, 7, 2, 1, 5, 3, 7, 8, 8, 0, 1, 3, 4, 3, 1, 9, 7, 4, 6, 7, 8, 5, 4, 7, 9, 0, 7, 3, 0, 7, 8, 1, 4, 3, 7, 4, 9, 4, 7, 2, 6, 1, 4, 3, 9, 4, 4, 8, 7, 3, 2, 6, 4, 6, 3, 1, 6, 4, 6, 9, 2, 5, 6, 4, 3, 0, 0, 8, 6, 4, 1, 6, 0, 4, 6, 2, 5, 2, 7, 5, 9, 5, 4, 2, 9, 3, 4, 5, 6

Offset: 1

Jean-François Alcover, Apr 14 2014

			1.2755477364172...

Steven Finch, Pattern-Avoiding Permutations [Internet archive]
Steven Finch, Pattern-Avoiding Permutations [Cached copy, with permission]

Cf. A052319, A071075, A111004, A200404, A263885.

RealDigits[Sqrt[2]*InverseErf[Sqrt[2/Pi]], 10, 100] // First

a=sqrt(Pi/2); b=a-1; c=1/sqrt(2); solve(x=1,2, a*erfc(c*x)-b) \\ Charles R Greathouse IV, Sep 02 2024

Solution to sqrt(Pi/2)*erf(x/sqrt(2)) = 1.

A197365 T(n,k) gives the number of permutations of the set [n] that contain k occurrences of the subword (132); irregular array read by rows (n >= 0 and 0 <= k <= max(0, floor((n-1)/2))).

Original entry on oeis.org

1, 1, 2, 5, 1, 16, 8, 63, 54, 3, 296, 368, 56, 1623, 2649, 753, 15, 10176, 20544, 9024, 576, 71793, 172596, 104814, 13572, 105, 562848, 1569408, 1228608, 259968, 7968, 4853949, 15398829, 14824314, 4532034, 306729, 945, 45664896, 162412416, 185991936, 75929856

Offset: 0

Peter Bala, Oct 14 2011

A permutation p(1)p(2)...p(n) in the symmetric group S_n contains the subword (132) if there are 3 consecutive elements p(i)p(j)p(k) that have the same order relations as (132), that is, p(i) < p(j) > p(k) and p(i) < p(k). For the enumeration of permutations containing the subword (123) see A162975.
From Petros Hadjicostas, Nov 05 2019: (Start)
The attached Maple program gives a recurrence for the o.g.f. of each row in terms of u for T(n,k), the number of permutations of [n] containing exactly k occurrences of the consecutive pattern 123...(r+1)(r+3)(r+2) for r >= 0. In the program, t = r + 2. Here, n >= 0 and 0 <= k <= max(0, (n-1)/t).
Using that recurrence we may get any row or column from the irregular triangular array T(n, k) for any r >= 0. (Here r = 0, while in array A264781 we have r = 2.)
The recurrence follows from manipulation of the bivariate o.g.f/e.g.f. 1/W(u,z) = Sum_{n, k >= 0} T(n, k)*u^k*z^n/n!, whose reciprocal W(u,z) is the solution of the o.d.e. in Theorem 3.2 in Elizalde and Noy (2003) (with m = a = r + 1). The number t = r + 2 is the order of the o.d.e. in terms of the variable z.
(End)

			Table begins
.n\k.|......0......1.....2......3
= = = = = = = = = = = = = = = = =
..0..|......1
..1..|......1
..2..|......2
..3..|......5......1
..4..|.....16......8
..5..|.....63.....54.....3
..6..|....296....368....56
..7..|...1623...2649...753....15
..8..|..10176..20544..9024...576
...
T(4,0) = 16: The 16 permutations of S_4 not containing the subword (132) are (1234), (2134), (2314), (3124), (3214), (1342), (2341), (3241), (2413), (3412), (3421), (4123), (4213), (4231), (4312), (4321).
T(4,1) = 8: The 8 permutations of S_4 with 1 occurrence of the subword (132) are 1243, 1324, 1423, 1432, 2143, 2431, 3142, 4132.

Alois P. Heinz, Rows n = 0..100, flattened
A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
S. Elizalde and M. Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125.
Petros Hadjicostas, General Maple program for a recurrence for the number of permutations of [n] containing exactly k times the consecutive pattern 123...(r+1)(r+3)(r+2) for r >= 0. [Here r = 0 and in the program t = r + 2 = 2.]

Columns k=0-10 give: A111004, A263885, A263886, A263887, A263888, A263889, A263890, A263891, A263892, A263893, A263894.
T(2n+1,n) gives A001147.
T(2n+2,n) gives 2*A076729.
Cf. A162975, A264781 (pattern 12354).

b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
      add(b(u-j, o+j-1, 0)*`if`(j (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);  # Alois P. Heinz, Oct 30 2013

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, 0]*If[jJean-François Alcover, Mar 05 2015, after Alois P. Heinz *)

E.g.f.: 1/(1 - int_{t = 0..z} exp((u-1)*t^2/2!)) = sqrt(1 - u)/(sqrt(1 - u) -sqrt(Pi/2) * erf(z/2*sqrt(1 - u))) = 1 + z + 2*z^2/2! + (5 + u)*z^3/3! + (16 + 8*u)*z^4/4! + ....

n-th row sum = n!. First column is A111004.

A071075 Number of permutations that avoid the generalized pattern 132-4.

Original entry on oeis.org

1, 1, 2, 6, 23, 107, 585, 3671, 25986, 204738, 1776327, 16824237, 172701135, 1909624371, 22626612450, 285982186662, 3840440707485, 54603776221965, 819424594880559, 12942757989763101, 214626518776190178, 3728112755679416898, 67692934780306842501, 1282399636333412178531, 25303124674163685176793

Offset: 0

Sergey Kitaev, May 26 2002

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..460
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019. See Table 1.
Andrew M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter, Algorithms for Permutation Statistics, Ph. D. Dissertation, Rutgers University, May 2011.
Andrew M. Baxter and Lara K. Pudwell, Enumeration schemes for dashed patterns, arXiv:1108.2642 [math.CO], 2011.
Sergey Kitaev, Partially Ordered Generalized Patterns, preprint.
Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.
Yan Wang, Qi Fang, Shishuo Fu, Sergey Kitaev, and Haijun Li, Consecutive and quasi-consecutive patterns: des-Wilf classifications and generating functions, arXiv:2502.10128 [math.CO], 2025. See p. 6.

Cf. A071088, A071076, A071077, A111004.

A(y) := 1/(1-int(exp(-t^2/2),t=0..y)); B(x) := exp(int(A(y),y=0..x)); series(B(x),x=0,30);

CoefficientList[Series[E^(Integrate[1/(1-Integrate[E^(-t^2/2), {t,0,y}]), {y,0,x}]), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 23 2014 *)

N=66; x='x+O('x^N);
A=1/(1-intformal(exp(-x^2/2)));
egf=exp(intformal(A));
Vec(serlaplace(egf))
\\ Joerg Arndt, Aug 28 2014

E.g.f.: exp(int(A(y), y=0..x)), where A(y) = 1/(1 - int(exp(-t^2/2), t=0..y)).
a(n) ~ c * d^n * n! / n^f, where d = 1/A240885 = 1/(sqrt(2)*InverseErf(sqrt(2/Pi))) = 0.7839769312035474991242486548698125357473282..., f = 1.2558142944089303287268746534354522944538722816671534535062816..., c = 0.2242410644782853722452053227678681810005068... . - Vaclav Kotesovec, Aug 23 2014
Let b(n) = A111004(n) = number of permutations of [n] that avoid the consecutive pattern 132. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i)*b(i)*a(n-1-i) with a(0) = b(0) = 1. [See the recurrence for A_n and B_n in the proof of Theorem 13 in Kitaev's papers.] - Petros Hadjicostas, Nov 01 2019

Link and a(11)-a(20) from Andrew Baxter, May 17 2011

Typo in first formula corrected by Vaclav Kotesovec, Aug 23 2014

A327722 Number T(m,n) of permutations of [n] avoiding the consecutive pattern 12...(m+1)(m+3)(m+2), where m, n >= 0; array read by ascending antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 5, 1, 1, 2, 6, 16, 1, 1, 2, 6, 23, 63, 1, 1, 2, 6, 24, 110, 296, 1, 1, 2, 6, 24, 119, 630, 1623, 1, 1, 2, 6, 24, 120, 708, 4204, 10176, 1, 1, 2, 6, 24, 120, 719, 4914, 32054, 71793, 1, 1, 2, 6, 24, 120, 720, 5026, 38976, 274914, 562848

Offset: 0

Petros Hadjicostas, Nov 02 2019

By taking complements of permutations, we see that T(m,n) is also the number of permutations of [n] avoiding the consecutive pattern (m+3)(m+2)...(3)(1)(2). [The complement of permutation (c_1,c_2,...,c_n) of [n] is (n + 1 - c_1, n + 1 - c_2, ..., n + 1 - c_n).]
If we let S(n,k) = T(n-k, k) for n >= 0 and 0 <= k <= n, we get a triangular array shown in the Example section below.
Note that lim_{n -> oo} S(n,k) = k! = A000142(k) for k >= 0.
By using the ratio test and the Stirling approximation to the Gamma function, we may show that the radius of convergence of the power series W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(b(n, m+2)*((m + 2)*n + 1)) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the power series) is entire.

			Array T(m, n) (with rows m >= 0 and columns n >= 0) begins as follows:
  1, 1, 2, 5, 16,  63, 296, 1623, 10176,  71793, ...
  1, 1, 2, 6, 23, 110, 630, 4204, 32054, 274914, ...
  1, 1, 2, 6, 24, 119, 708, 4914, 38976, 347765, ...
  1, 1, 2, 6, 24, 120, 719, 5026, 40152, 360864, ...
  1, 1, 2, 6, 24, 120, 720, 5039, 40304, 362664, ...
  1, 1, 2, 6, 24, 120, 720, 5040, 40319, 362862, ...
  ...
Triangular array S(n, k) = T(n-k, k) (with rows n >= 0 and columns k >= 0) begins as follows:
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 5;
  1, 1, 2, 6, 16;
  1, 1, 2, 6, 23,  63;
  1, 1, 2, 6, 24, 110, 296;
  1, 1, 2, 6, 24, 119, 630, 1623;
  1, 1, 2, 6, 24, 120, 708, 4204, 10176;
  1, 1, 2, 6, 24, 120, 719, 4914, 32054,  71793;
  1, 1, 2, 6, 24, 120, 720, 5026, 38976, 274914, 562848;
  ...

A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes, 2011.
Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116) with u = 0 and m and a in the theorem equal to our m + 1.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Rows include A111004 (m = 0, pattern 132), A117226 (m = 1, pattern 1243), A202213 (m = 2, pattern 12354).

Cf. A000142, A033312, A242569, A329070.

E.g.f for row m >= 0: 1/W_m(z), where W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(b(n, m+2)*((m + 2)*n + 1)) with b(n, k) = A329070(n, k) = (k*n)!/(k^n * (1/k)_n). (Here (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol, or rising factorial, which is denoted by (x)^n in some papers and books.)
The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).
T(m, n) = Sum_{s = 0..floor((n - 1)/(m + 2))} (-(m + 2))^s * (1/(m + 2))_s * binomial(n, (m + 2)*s + 1) * T(m, n - (m + 2)*s - 1) for n >= 1 with T(m, 0) = 1.
T(m, n) = n! for 0 <= n <= m + 2.
T(m, m+3) = (m + 3)! - 1 = A000142(m + 3) - 1 = A033312(m + 3) for m >= 0. [In the set of permutations of [m + 3] there is exactly one permutation that contains the pattern 12...(m+1)(m+3)(m+2).]
Conjecture: T(m, m + 4) = A242569(m + 4) = (m + 4)! - 2*(m + 4) for m >= 0.
Limit_{m -> oo} T(m, n) = n! = A000142(n) for n >= 0.

A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 8, 6, 1, 1, 48, 180, 24, 1, 1, 384, 12960, 8064, 120, 1, 1, 3840, 1710720, 10644480, 604800, 720, 1, 1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1, 1, 645120, 109930867200, 244635697152000, 2303884477440000, 70355755008000, 10897286400, 40320, 1

Offset: 0

Petros Hadjicostas, Nov 03 2019

For information about the function W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)) (mentioned in the Formula section below), see Theorem 3.2 in Elizalde and Noy (2003) with u = 0 and m and a in the theorem equal to our m + 1. See also the documentation of array A327722.
By using the ratio test and the Stirling approximation to the gamma function, we may show that the radius of convergence of the power series for W_m(z) is infinity (for each m >= 0). Thus, the function W_m(z) (as defined by the above power series) is entire.
If we define S(m,s) = T(n-s, s+1) for m >= 0 and 0 <= s <= m, we get the triangular array that appears in the Example section below.

			Array T(n,k) (with rows n >= 0 and columns k >= 1) begins as follows:
  1,  1,     1,        1,           1,              1,  ...
  1,  2,     6,       24,         120,            720,  ...
  1,  8,   180,     8064,      604800,       68428800, ...
  1, 48, 12960, 10644480, 19813248000, 70355755008000, ...
  ...
Triangular array S(m,s) = T(m-s, s+1) (with rows m >= 0 and columns s >= 0):
  1;
  1,     1;
  1,     2,         1;
  1,     8,         6,           1;
  1,    48,       180,          24,           1;
  1,   384,     12960,        8064,         120,        1;
  1,  3840,   1710720,    10644480,      604800,      720,    1;
  1, 46080, 359251200, 35765452800, 19813248000, 68428800, 5040, 1;
  ...

Sergi Elizalde and Marc Noy, Consecutive patterns in permutations, Adv. Appl. Math. 30 (2003), 110-125; see Theorem 3.2 (p. 116).
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
Eric Weisstein's World of Mathematics, Pochhammer Symbol.
Wikipedia, Falling and rising factorials.

Rows include A000012 (n = 0), A000142 (n = 1), A060593 (n = 2).
Columns include A000012 (k = 1), A000165 (k = 2), A176730 (k = 3).
Ratios T(n+1,k)/(k!*T(n,k)) include A000012 (k = 1), A000027 (k = 2), A000326 (k = 3), A100157 (k = 4), A234043 (k = 5).
Cf. A111004, A117226, A202213, A327722.

A := (n, k) -> `if`(k=0, 1, (GAMMA(1/k)*GAMMA(k*n+1))/(GAMMA(n+1/k)*k^n)):
seq(seq(A(n-k-1, k), k=1..n-1), n=0..10); # Peter Luschny, Nov 04 2019

T(0,k) = 1, T(1,k) = k!, and T(2,k) = (2*k)!/(k + 1) for k >= 1.
T(n,1) = 1, T(n,2) = (2*n)!!, and T(n,3) is related to the Airy functions (see the documentation of A176730).
T(n+1,k) = (k-1)! * binomial(k*(n+1), k-1) * T(n,k) for n >= 0 and k >= 1.
T(n+1,k)/(k! * T(n,k)) = Cat(n+1, k), where Cat(d, k) = binomial(k*d, k)/(k * (d - 1) + 1) is a Fuss-Catalan number; see Theorem 1.2 in Schuetz and Whieldon (2014).
If F(k,z) = Sum_{n >= 0} z^(k*n)/T(n,k), then F(k,z) satisfies the o.d.e. F^(k-1)(k,z) - z*F(k,z) = 0.
If W_m(z) = 1 + Sum_{n >= 0} (-1)^(n+1)* z^((m+2)*n + 1)/(T(n, m+2)*((m + 2)*n + 1)), then 1/W_m(z) is the e.g.f. of row m of A327722(m,n), which counts permutations of [n] that avoid the consecutive pattern 12...(m+1)(m+3)(m+2) (or equivalently, the consecutive pattern (m+3)(m+2)...(3)(1)(2)).
The function W_m(z) satisfies the o.d.e. W_m^(m+2)(z) + z*W_m'(z) = 0 with W_m(0) = 1, W_m'(0) = -1, and W_m^(s)(0) = 0 for s = 2..(m + 1).

A324130 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 1 and t = 132.

Original entry on oeis.org

1, 1, 2, 6, 24, 116, 652, 4178, 30070, 240164, 2107606, 20156458, 208639514, 2323794632, 27709659880, 352203163790, 4753474785808, 67889631514128, 1022936113573148, 16216615869916570, 269816176058513398, 4701111255106851632, 85599799432794431978, 1625828159969984754538

Offset: 0

N. J. A. Sloane, Feb 16 2019

nonn

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Cf. A000142, A111004.

From Petros Hadjicostas, Oct 29 2019: (Start)
Let b(n) = A111004(n) = number of permutations avoiding a consecutive 132 pattern. Then a(n) = 2*a(n-1) - b(n-1) + Sum_{i = 1..n-1} binomial(n-1,i) * b(i) * a(n-1-i) for n >= 1 with a(0) = b(0) = 1. [See the recurrence for C_n on p. 220 of Kitaev (2005).]
E.g.f.: If A(x) is the e.g.f. of (a(n): n >= 0) and B(x) is the e.g.f. of (b(n): n >= 0) (i.e., B(x) = 1/(1 - Int(exp(-t^2/2), t = 0..x))), then A'(x) = (1 + B(x)) * A(x) - B(x) with A(0) = B(0) = 1. [Theorem 16, p. 219, in Kitaev (2005)] (End)

More terms from Petros Hadjicostas, Oct 29 2019 using a recurrence by Kitaev (2005)

A324134 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 12 and t = 132.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 710, 4800, 36298, 302780, 2758618, 27246450, 289962508, 3308024082, 40278949800, 521427324542, 7152011191362, 103621538280688, 1581465201545374, 25361207137790358, 426374509273382756, 7499269147438400178, 137728268057069904088, 2636572230825216681414

Offset: 0

N. J. A. Sloane, Feb 16 2019

nonn

			From _Petros Hadjicostas_, Oct 31 2019: (Start)
In a permutation of [n] that contains the shuffle pattern s-k-t, where s = 12 and t = 132, k should be greater than the numbers in pattern s and the numbers in pattern t. (The numbers in each of the patterns s and t should be contiguous.) Clearly, for n = 0..5, all permutations of [n] avoid this shuffle pattern (since we need at least six numbers to get this pattern). Hence, a(n) = n! for n = 0..5.
For n = 6, k should be equal to 6, and for the pattern s = 12 we have the 10 choices 12, 13, 14, 15, 23, 24, 25, 34, 35, and 45. The corresponding permutations of [6] that contain this shuffle pattern are 126354, 136254, 146253, 156243, 236154, 246153, 256143, 346152, 356142, and 456132. Thus, a(6) = 6! - 10 = 710. (End)

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Cf. A000142, A111004, A324130.

Let b(n) = A111004(n) = number of permutations avoiding a consecutive 132 pattern. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i) * (a(n-1-i) + b(i) * a(n-1-i) - b(n-1-i)) for n >= 1 with a(0) = b(0) = 1. [See the recurrence for C_n on p. 220 of Kitaev (2005).] - Petros Hadjicostas, Oct 30 2019

More terms from Petros Hadjicostas, Oct 30 2019

A324138 Number of permutations of [n] that avoid the shuffle pattern s-k-t, where s = 123 and t = 132.

Original entry on oeis.org

1, 1, 2, 6, 24, 120, 720, 5020, 39755, 351518, 3425572, 36419844, 419026188, 5182797757, 68535001302, 964404124479, 14383519018582, 226579159065496, 3758349089828472, 65466833442028670, 1194655878120996337, 22788580047064423474, 453513206778006345040

Offset: 0

N. J. A. Sloane, Feb 16 2019

nonn

			From _Petros Hadjicostas_, Nov 01 2019: (Start)
In a permutation of [n] that contains the shuffle pattern s-k-t, where s = 123 and t = 132, k should be greater than the numbers in pattern s and the numbers in pattern t. (The numbers in each of the patterns s and t should be contiguous.) Clearly, for n = 0..6, all permutations of [n] avoid this shuffle pattern (since we need at least seven numbers to get this pattern). Hence, a(n) = n! for n = 0..6.
For n = 7, k should be equal to 7, and for the pattern s = 123 we have binomial(6,3) = 20 choices: 123, 124, 125, ..., 456. The corresponding permutations of [7] that contain this shuffle pattern are 1237465, 1247365, 1257364, ..., 4567132. Thus, a(7) = 7! - 20 = 5020. (End)

Sergey Kitaev, Partially Ordered Generalized Patterns, Discrete Math. 298 (2005), no. 1-3, 212-229.

Cf. A000142, A049774, A111004.

Let b(n) = A049774(n) = number of permutations of [n] that avoid consecutive pattern s = 123 and c(n) = A111004(n) = number of permutations of [n] that avoid consecutive pattern t = 132. Then a(n) = Sum_{i = 0..n-1} binomial(n-1,i) * (b(i)*a(n-1-i) + c(i)*a(n-1-i) - b(i)*c(n-1-i)) for n >= 1 with a(0) = b(0) = c(0) = 1. [This follows from the recurrence for C_n on p. 220 in Kitaev (2005).] - Petros Hadjicostas, Nov 01 2019

More terms from Petros Hadjicostas, Nov 01 2019

cp's OEIS Frontend

A049774 Number of permutations of n elements not containing the consecutive pattern 123.

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A117158 Number of permutations avoiding the consecutive pattern 1234.

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A240885 Decimal expansion of the unique positive solution of Integral_{0..x} exp(-t^2/2) dt = 1.

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A197365 T(n,k) gives the number of permutations of the set [n] that contain k occurrences of the subword (132); irregular array read by rows (n >= 0 and 0 <= k <= max(0, floor((n-1)/2))).

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A071075 Number of permutations that avoid the generalized pattern 132-4.

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A327722 Number T(m,n) of permutations of [n] avoiding the consecutive pattern 12...(m+1)(m+3)(m+2), where m, n >= 0; array read by ascending antidiagonals.

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A329070 Array read by ascending antidiagonals: T(n, k) = (k*n)!/(k^n*(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x*(x + 1)*...*(x + n - 1) is the Pochhammer symbol.

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A329070 Array read by ascending antidiagonals: T(n, k) = (kn)!/(k^n(1/k)_n) with (n >= 0 and k >= 1), where (x)_n = x(x + 1)...*(x + n - 1) is the Pochhammer symbol.