A226316 -id:A226316 - cp's OEIS frontend

A102726 Number of compositions of the integer n into positive parts that avoid a fixed pattern of three letters.

1, 1, 2, 4, 8, 16, 31, 60, 114, 214, 398, 732, 1334, 2410, 4321, 7688, 13590, 23869, 41686, 72405, 125144, 215286, 368778, 629156, 1069396, 1811336, 3058130, 5147484, 8639976, 14463901, 24154348, 40244877, 66911558, 111026746, 183886685, 304034456, 501877227

Offset: 0

Herbert S. Wilf, Feb 07 2005

The sequence is the same no matter which of the six patterns of three letters is chosen as the one to be avoided.

			a(6) = 31 because there are 32 compositions of 6 into positive parts and only one of these, namely 6 = 1+2+3, contains the pattern (123), the other 31 compositions of 6 avoid that pattern.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..900 (first 400 terms from Alois P. Heinz)
M. Elder and V. Vatter, Problems and conjectures presented at the third international conference on permutation petterns
Carla D. Savage and Herbert S. Wilf, Pattern avoidance in compositions and multiset permutations, Advances in Applied Mathematics 36 (2006), pp.194-201.
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for patterns is A226316.
These compositions are ranked by the complement of A335479.
The matching version is A335514.
The version for prime indices is A335521.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Compositions are counted by A011782.
Strict compositions are counted by A032020 and ranked by A233564.
Patterns matched by compositions are counted by A335456.
Minimal patterns avoided by a given composition are counted by A335465.
Cf. A056986, A106356, A232464, A269134, A333755.

b:= proc(n, m, t) option remember; `if`(n=0, 1,
      add(b(n-i, min(m, i, n-i), min(t, n-i,
      `if`(i>m, i, t))), i=1..min(n, t)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..50);  # Alois P. Heinz, Mar 18 2014

b[n_, m_, t_] := b[n, m, t] = If[n == 0, 1, Sum[b[n - i, Min[m, i, n - i], Min[t, n - i, If[i > m, i, t]]], {i, 1, Min[n, t]}]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 10 2017, after Alois P. Heinz *)
mstype[q_]:=q/.Table[Union[q][[i]]->i,{i,Length[Union[q]]}];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],!MemberQ[Union[mstype/@Subsets[#]],{1,2,3}]&]],{n,0,10}] (* Gus Wiseman, Jun 22 2020 *)

seq(n)={Vec(sum(i=1, n, prod(j=1, n, if(i==j, 1, (1-x^i)/((1-x^(j-i))*(1-x^i-x^j))) + O(x*x^n))/(1-x^i)))} \\ Andrew Howroyd, Dec 31 2020

G.f.: Sum_{i>=1} (1/(1-x^i))*Product_{j>=1, j<>i} (1-x^i)/((1-x^(j-i))*(1-x^i-x^j)).

Asymptotics (Savage and Wilf, 2005): a(n) ~ c * ((1+sqrt(5))/2)^n, where c = r/(r-1)/(r-s) * (r * Product_{j>=3} (1-1/r)/(1-r^(1-j))/(1-1/r-r^(-j)) - Product_{j>=3} (1-1/r^2)/(1-r^(2-j))/(1-1/r^2-r^(-j)) ) = 18.9399867283479198666671671745270505487677312850521421513193261105... and r = (1+sqrt(5))/2, s = (1-sqrt(5))/2. - Vaclav Kotesovec, May 02 2014

More terms from Ralf Stephan, May 27 2005

A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 6, 6, 0, 1, 14, 36, 24, 0, 1, 30, 150, 240, 120, 0, 1, 62, 540, 1560, 1800, 720, 0, 1, 126, 1806, 8400, 16800, 15120, 5040, 0, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 0, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880

Offset: 0

Philippe Deléham, Sep 14 2007

Triangle T(n,k), 0 <= k <= n, read by rows given by [0,1,0,2,0,3,0,4,0,5,0,6,0,7,0,...] DELTA [1,1,2,2,3,3,4,4,5,5,6,6,...] where DELTA is the operator defined in A084938; another version of A019538.
See also A019538: version with n > 0 and k > 0. - Philippe Deléham, Nov 03 2008
From Peter Bala, Jul 21 2014: (Start)
T(n,k) gives the number of (k-1)-dimensional faces in the interior of the first barycentric subdivision of the standard (n-1)-dimensional simplex. For example, the barycentric subdivision of the 1-simplex is o--o--o, with 1 interior vertex and 2 interior edges, giving T(2,1) = 1 and T(2,2) = 2.
This triangle is used when calculating the face vectors of the barycentric subdivision of a simplicial complex. Let S be an n-dimensional simplicial complex and write f_k for the number of k-dimensional faces of S, with the usual convention that f_(-1) = 1, so that F := (f_(-1), f_0, f_1,...,f_n) is the f-vector of S. If M(n) denotes the square matrix formed from the first n+1 rows and n+1 columns of the present triangle, then the vector F*M(n) is the f-vector of the first barycentric subdivision of the simplicial complex S (Brenti and Welker, Lemma 2.1). For example, the rows of Pascal's triangle A007318 (but with row and column indexing starting at -1) are the f-vectors for the standard n-simplexes. It follows that A007318*A131689, which equals A028246, is the array of f-vectors of the first barycentric subdivision of standard n-simplexes. (End)
This triangle T(n, k) appears in the o.g.f. G(n, x) = Sum_{m>=0} S(n, m)*x^m with S(n, m) = Sum_{j=0..m} j^n for n >= 1 as G(n, x) = Sum_{k=1..n} (x^k/(1 - x)^(k+2))*T(n, k). See also the Eulerian triangle A008292 with a Mar 31 2017 comment for a rewritten form. For the e.g.f. see A028246 with a Mar 13 2017 comment. - Wolfdieter Lang, Mar 31 2017
T(n,k) = the number of alignments of length k of n strings each of length 1. See Slowinski. An example is given below. Cf. A122193 (alignments of strings of length 2) and A299041 (alignments of strings of length 3). - Peter Bala, Feb 04 2018
The row polynomials R(n,x) are the Fubini polynomials. - Emanuele Munarini, Dec 05 2020
From Gus Wiseman, Feb 18 2022: (Start)
Also the number of patterns of length n with k distinct parts (or with maximum part k), where we define a pattern to be a finite sequence covering an initial interval of positive integers. For example, row n = 3 counts the following patterns:
  (1,1,1)  (1,2,2)  (1,2,3)
           (2,1,2)  (1,3,2)
           (2,2,1)  (2,1,3)
           (1,1,2)  (2,3,1)
           (1,2,1)  (3,1,2)
           (2,1,1)  (3,2,1)
(End)
Regard A048994 as a lower-triangular matrix and divide each term A048994(n,k) by n!, then this is the matrix inverse. Because Sum_{k=0..n} (A048994(n,k) * x^n / n!) = A007318(x,n), Sum_{k=0..n} (A131689(n,k) * A007318(x,k)) = x^n. - Natalia L. Skirrow, Mar 23 2023
T(n,k) is the number of ordered partitions of [n] into k blocks. - Alois P. Heinz, Feb 21 2025

			The triangle T(n,k) begins:
  n\k 0 1    2     3      4       5        6        7        8        9      10 ...
  0:  1
  1:  0 1
  2:  0 1    2
  3:  0 1    6     6
  4:  0 1   14    36     24
  5:  0 1   30   150    240     120
  6:  0 1   62   540   1560    1800      720
  7:  0 1  126  1806   8400   16800    15120     5040
  8:  0 1  254  5796  40824  126000   191520   141120    40320
  9:  0 1  510 18150 186480  834120  1905120  2328480  1451520   362880
  10: 0 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
  ... reformatted and extended. - _Wolfdieter Lang_, Mar 31 2017
From _Peter Bala_, Feb 04 2018: (Start)
T(4,2) = 14 alignments of length 2 of 4 strings of length 1. Examples include
  (i) A -    (ii) A -    (iii) A -
      B -         B -          - B
      C -         - C          - C
      - D         - D          - D
There are C(4,1) = 4 alignments of type (i) with a single gap character - in column 1, C(4,2) = 6 alignments of type (ii) with two gap characters in column 1 and C(4,3) = 4 alignments of type (iii) with three gap characters in column 1, giving a total of 4 + 6 + 4 = 14 alignments. (End)

Vincenzo Librandi, Rows n = 0..100, flattened
Peter Bala, Deformations of the Hadamard product of power series
F. Brenti and V. Welker, f-vectors of barycentric subdivisions, arXiv:math/0606356 [math.CO], Math. Z., 259(4), 849-865, 2008.
M. Dukes and C. D. White, Web Matrices: Structural Properties and Generating Combinatorial Identities, arXiv:1603.01589 [math.CO], 2016.
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 31-41.
Jerry Metzger and Thomas Richards, A Prisoner Problem Variation, Journal of Integer Sequences, Vol. 18 (2015), Article 15.2.7.
Massimo Nocentini, An algebraic and combinatorial study of some infinite sequences of numbers supported by symbolic and logic computation, PhD Thesis, University of Florence, 2019. See Ex. 36.
Mircea Dan Rus, Yet another note on notation, arXiv:2501.08762 [math.HO], 2025. See p. 6.
J. B. Slowinski, The Number of Multiple Alignments, Molecular Phylogenetics and Evolution 10:2 (1998), 264-266. doi:10.1006/mpev.1998.0522
M. Z. Spivey, On Solutions to a General Combinatorial Recurrence, J. Int. Seq. 14 (2011) # 11.9.7.
Wikipedia, Barycentric subdivision
Wikipedia, Simplicial complex
Wikipedia, Simplex
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Columns k=0..10 are A000007, A000012, A000918, A001117, A000919, A001118, A000920, A135456, A133068, A133360, A133132,
Case m=1 of the polynomials defined in A278073.
Cf. A000142 (diagonal), A000670 (row sums), A000012 (alternating row sums), A210029 (central terms).
Cf. A001286, A037960, A037961, A037962, A037963.
Cf. A008292, A028246 (o.g.f. and e.g.f. of sums of powers).
Cf. A019538, A122193, A299041.
A version for partitions is A116608, or by maximum A008284.
A version for compositions is A235998, or by maximum A048004.
Classes of patterns:
- A000142 = strict
- A005649 = anti-run, complement A069321
- A019536 = necklace
- A032011 = distinct multiplicities
- A060223 = Lyndon
- A226316 = (1,2,3)-avoiding, weakly A052709, complement A335515
- A296975 = aperiodic
- A345194 = alternating, up/down A350354, complement A350252
- A349058 = weakly alternating
- A351200 = distinct runs
- A351292 = distinct run-lengths
Cf. A007318, A097805, A335456, A335457, A344605, A349050.

function T(n, k)
    if k < 0 || k > n return 0 end
    if n == 0 && k == 0 return 1 end
    k*(T(n-1, k-1) + T(n-1, k))
end
for n in 0:7
    println([T(n, k) for k in 0:n])
end
# Peter Luschny, Mar 26 2020

A131689 := (n,k) -> Stirling2(n,k)*k!: # Peter Luschny, Sep 17 2011
# Alternatively:
A131689_row := proc(n) 1/(1-t*(exp(x)-1)); expand(series(%,x,n+1)); n!*coeff(%,x,n); PolynomialTools:-CoefficientList(%,t) end:
for n from 0 to 9 do A131689_row(n) od; # Peter Luschny, Jan 23 2017

t[n_, k_] := k!*StirlingS2[n, k]; Table[t[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 25 2014 *)
T[n_, k_] := If[n <= 0 || k <= 0, Boole[n == 0 && k == 0], Sum[(-1)^(i + k) Binomial[k, i] i^(n + k), {i, 0, k}]]; (* Michael Somos, Jul 08 2018 *)

{T(n, k) = if( n<0, 0, sum(i=0, k, (-1)^(k + i) * binomial(k, i) * i^n))};
/* Michael Somos, Jul 08 2018 */

@cached_function
def F(n): # Fubini polynomial
    R. = PolynomialRing(ZZ)
    if n == 0: return R(1)
    return R(sum(binomial(n, k)*F(n - k)*x for k in (1..n)))
for n in (0..9): print(F(n).list()) # Peter Luschny, May 21 2021

T(n,k) = k*(T(n-1,k-1) + T(n-1,k)) with T(0,0)=1. Sum_{k=0..n} T(n,k)*x^k = (-1)^n*A000629(n), A033999(n), A000007(n), A000670(n), A004123(n+1), A032033(n), A094417(n), A094418(n), A094419(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6 respectively. [corrected by Philippe Deléham, Feb 11 2013]
Sum_{k=0..n} T(n,k)*x^(n-k) = A000012(n), A000142(n), A000670(n), A122704(n) for x=-1, 0, 1, 2 respectively. - Philippe Deléham, Oct 09 2007
Sum_{k=0..n} (-1)^k*T(n,k)/(k+1) = Bernoulli numbers A027641(n)/A027642(n). - Peter Luschny, Sep 17 2011
G.f.: F(x,t) = 1 + x*t + (x+x^2)*t^2/2! + (x+6*x^2+6*x^3)*t^3/3! + ... = Sum_{n>=0} R(n,x)*t^n/n!.
The row polynomials R(n,x) satisfy the recursion R(n+1,x) = (x+x^2)*R'(n,x) + x*R(n,x) where ' indicates differentiation with respect to x. - Philippe Deléham, Feb 11 2013
T(n,k) = [t^k] (n! [x^n] (1/(1-t*(exp(x)-1)))). - Peter Luschny, Jan 23 2017
The n-th row polynomial has the form x o x o ... o x (n factors), where o denotes the black diamond multiplication operator of Dukes and White. See also Bala, Example E8. - Peter Bala, Jan 08 2018

A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

Original entry on oeis.org

0, 1, 1, 3, 9, 31, 113, 431, 1697, 6847, 28161, 117631, 497665, 2128127, 9183489, 39940863, 174897665, 770452479, 3411959809, 15181264895, 67833868289, 304256253951, 1369404661761, 6182858317823, 27995941060609, 127100310290431, 578433619525633, 2638370120138751

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

A simple context-free grammar.
Number of lattice paths from (0,0) to (2n-2,0) that stay (weakly) in the first quadrant and such that each step is either U=(1,1), D=(1,-1), or L=(3,1). Equivalently, underdiagonal lattice paths from (0,0) to (n-1,n-1) and such that each step is either (1,0), (0,1), or (2,1). E.g., a(4)=9 because in addition to the five Dyck paths from (0,0) to (6,0) [UDUDUD, UDUUDD, UUDDUD, UUDUDD, UUUDDD] we have LDUD, LUDD, ULDD and UDLD. - Emeric Deutsch, Dec 21 2003
Hankel transform of a(n+1) is A006125(n+1). - Paul Barry, Apr 01 2007
Also, a(n+1) is the number of walks from (0,0) to (n,0) using steps (1,1), (1,-1) and (0,-1). See the U(n,k) array in A071943, where A052709(n+1) = U(n,0). - N. J. A. Sloane, Mar 29 2013
Diagonal sums of triangle in A085880. - Philippe Deléham, Nov 15 2013
From Gus Wiseman, Jun 17 2021: (Start)
Conjecture: For n > 0, also the number of sequences of length n - 1 covering an initial interval of positive integers and avoiding three terms (..., x, ..., y, ..., z, ...) such that x <= y <= z. The version avoiding the strict pattern (1,2,3) is A226316. Sequences covering an initial interval are counted by A000670. The a(1) = 1 through a(4) = 9 sequences are:
  ()  (1)  (1,1)  (1,2,1)
           (1,2)  (1,3,2)
           (2,1)  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (2,2,1)
                  (2,3,1)
                  (3,1,2)
                  (3,2,1)
(End)

N. J. A. Sloane, Table of n, a(n) for n = 0..499
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Ascending runs in permutations and valued Dyck paths, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 445-463.
Paul Barry, Riordan arrays, generalized Narayana triangles, and series reversion, Linear Algebra and its Applications, 491 (2016) 343-385.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Daniel Birmajer, Juan B. Gil, David S. Kenepp, and Michael D. Weiner, Restricted generating trees for weak orderings, arXiv:2108.04302 [math.CO], 2021.
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.
Xiang-Ke Chang, X.-B. Hu, H. Lei, and Y.-N. Yeh, Combinatorial proofs of addition formulas, The Electronic Journal of Combinatorics, 23(1) (2016), #P1.8.
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
M. Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
James East and Nicholas Ham, Lattice paths and submonoids of Z^2, arXiv:1811.05735 [math.CO], 2018.
L. Ferrari, E. Pergola, R. Pinzani, and S. Rinaldi, Jumping succession rules and their generating functions, Discrete Math., 271 (2003), 29-50.
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 664
J. P. S. Kung and A. de Mier, Catalan lattice paths with rook, bishop and spider steps, Journal of Combinatorial Theory, Series A 120 (2013) 379-389. - _N. J. A. Sloane_, Dec 27 2012
D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Diagonal entries of A071943 and A071945.
Cf. A000108, A000670, A025227, A052709, A056986, A071943, A085880.
Cf. A102726, A117434, A158005, A226316, A333217, A335479.

[0] cat [(&+[Binomial(n,k+1)*Binomial(2*k,n-1): k in [0..n-1]])/n: n in [1..30]]; // G. C. Greubel, May 30 2022

spec := [S,{C=Prod(B,Z),S=Union(B,C,Z),B=Prod(S,S)},unlabeled]: seq(combstruct[count](spec,size=n), n=0..20);

InverseSeries[Series[(y-y^2)/(1+y^2), {y, 0, 24}], x] (* then A(x)= y(x) *) (* Len Smiley, Apr 12 2000 *)
CoefficientList[Series[(1 -Sqrt[1 -4x -4x^2])/(2(1+x)), {x, 0, 33}], x] (* Vincenzo Librandi, Feb 12 2016 *)

a(n)=polcoeff((1-sqrt(1-4*x*(1+x+O(x^n))))/2/(1+x),n)

[sum(binomial(k, n-k-1)*catalan_number(k) for k in (0..n-1)) for n in (0..30)] # G. C. Greubel, May 30 2022

a(n) + a(n-1) = A025227(n).
a(n) = Sum_{k=0..floor((n-1)/2)} (2*n-2-2*k)!/(k!*(n-k)!*(n-1-2*k)!). - Emeric Deutsch, Nov 14 2001
D-finite with recurrence: n*a(n) = (3*n-6)*a(n-1) + (8*n-18)*a(n-2) + (4*n-12)*a(n-3), n>2. a(1)=a(2)=1.
a(n) = b(1)*a(n-1) + b(2)*a(n-2) + ... + b(n-1)*a(1) for n>1 where b(n)=A025227(n).
G.f.: A(x) = x/(1-(1+x)*A(x)). - Paul D. Hanna, Aug 16 2002
G.f.: A(x) = x/(1-z/(1-z/(1-z/(...)))) where z=x+x^2 (continued fraction). - Paul D. Hanna, Aug 16 2002; revised by Joerg Arndt, Mar 18 2011
a(n+1) = Sum_{k=0..n} Catalan(k)*binomial(k, n-k). - Paul Barry, Feb 22 2005
From Paul Barry, Mar 14 2006: (Start)
G.f. is x*c(x*(1+x)) where c(x) is the g.f. of A000108.
Row sums of A117434. (End)
a(n+1) = (1/(2*Pi))*Integral_{x=2-2*sqrt(2)..2+2*sqrt(2)} x^n*(4+4x-x^2)/(2*(1+x)). - Paul Barry, Apr 01 2007
From Gary W. Adamson, Jul 22 2011: (Start)
For n>0, a(n) is the upper left term in M^(n-1), where M is an infinite square production matrix as follows:
  1, 1, 0, 0, 0, 0, ...
  2, 1, 1, 0, 0, 0, ...
  2, 2, 1, 1, 0, 0, ...
  2, 2, 2, 1, 1, 0, ...
  2, 2, 2, 2, 1, 1, ...
  ... (End)
G.f.: x*Q(0), where Q(k) = 1 + (4*k+1)*x*(1+x)/(k+1 - x*(1+x)*(2*k+2)*(4*k+3)/(2*x*(1+x)*(4*k+3) + (2*k+3)/Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 14 2013
a(n) ~ sqrt(2-sqrt(2))*2^(n-1/2)*(1+sqrt(2))^(n-1)/(n^(3/2)*sqrt(Pi)). - Vaclav Kotesovec, Jun 29 2013
a(n+1) = Sum_{k=0..floor(n/2)} A085880(n-k,k). - Philippe Deléham, Nov 15 2013

Better g.f. and recurrence from Michael Somos, Aug 03 2000

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A344605 Number of alternating patterns of length n, including pairs (x,x).

Original entry on oeis.org

1, 1, 3, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, May 27 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670. A sequence is alternating (cf. A025047) including pairs (x,x) if there are no adjacent triples (..., x, y, z, ...) where x <= y <= z or x >= y >= z. These sequences avoid the weak consecutive patterns (1,2,3) and (3,2,1).
An alternating pattern of length > 2 is necessarily an anti-run (A005649).
The version without pairs (x,x) is identical to this sequence except a(2) = 2 instead of 3.

			The a(0) = 1 through a(4) = 22 patterns:
  ()  (1)  (1,1)  (1,2,1)  (1,2,1,2)
           (1,2)  (1,3,2)  (1,2,1,3)
           (2,1)  (2,1,2)  (1,3,1,2)
                  (2,1,3)  (1,3,2,3)
                  (2,3,1)  (1,3,2,4)
                  (3,1,2)  (1,4,2,3)
                           (2,1,2,1)
                           (2,1,3,1)
                           (2,1,3,2)
                           (2,1,4,3)
                           (2,3,1,2)
                           (2,3,1,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (3,1,2,1)
                           (3,1,3,2)
                           (3,1,4,2)
                           (3,2,3,1)
                           (3,2,4,1)
                           (3,4,1,2)
                           (4,1,3,2)
                           (4,2,3,1)

Martin Ehrenstein, Table of n, a(n) for n = 0..25

The version for permutations is A001250.
The version for compositions is A344604.
The version for permutations of prime indices is A344606.
A000670 counts patterns (ranked by A333217).
A003242 counts anti-run compositions.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A025047 counts alternating or wiggly compositions, complement A345192.
A226316 counts patterns avoiding (1,2,3) (weakly: A052709).
A335515 counts patterns matching (1,2,3).
Cf. A000041, A006330, A049774, A102726, A103919, A124754, A128761, A335456, A335457, A335517, A344612, A344614, A344615, A348377.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],!MatchQ[#,{_,x_,y_,z_,_}/;x<=y<=z||x>=y>=z]&]],{n,0,6}]

a(10) and beyond from Martin Ehrenstein, Jun 10 2021

A345194 Number of alternating patterns of length n.

Original entry on oeis.org

1, 1, 2, 6, 22, 102, 562, 3618, 26586, 219798, 2018686, 20393790, 224750298, 2683250082, 34498833434, 475237879950, 6983085189454, 109021986683046, 1802213242949602, 31447143854808378, 577609702827987882, 11139837273501641502, 225075546284489412854

Offset: 0

Gus Wiseman, Jun 17 2021

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
The version with twins (A344605) is identical to this sequence except with a(2) = 3 instead of 2.
From Gus Wiseman, Jan 16 2022: (Start)
Conjecture: Also the number of weakly up/down patterns of length n, where a sequence is weakly up/down if it is alternately weakly increasing and weakly decreasing, starting with an increase. For example, the a(0) = 1 through a(3) = 6 weakly up/down patterns are:
  ()  (1)  (1,1)  (1,1,1)
           (2,1)  (1,1,2)
                  (2,1,1)
                  (2,1,2)
                  (2,1,3)
                  (3,1,2)
(End)

			The a(0) = 1 through a(3) = 6 alternating patterns:
  ()  (1)  (1,2)  (1,2,1)
           (2,1)  (1,3,2)
                  (2,1,2)
                  (2,1,3)
                  (2,3,1)
                  (3,1,2)

Andrew Howroyd, Table of n, a(n) for n = 0..200

The version for permutations is A001250, complement A348615.
The version for compositions is A025047, complement A345192.
The version with twins (x,x) is A344605.
The version for perms of prime indices is A345164, complement A350251.
The version for factorizations is A348610, complement A348613, weak A349059.
The weak version is A349058, complement A350138, compositions A349052.
The complement is counted by A350252.
A000670 = patterns, ranked by A333217.
A003242 = anti-run compositions.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A129852 and A129853 = up/down and down/up compositions.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A006330, A049774, A128761, A335456, A335457, A335517, A344614, A344615, A345167, A345168.

wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]== Length[y]&&Length[Split[Sign[Differences[y]]]]==Length[y]-1];
allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],wigQ]],{n,0,6}]

F(p,x) = {sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k)}
R(n,k) = {Vec(if(k==1, x, 2*F(k-2,-x)/F(k-1,x)-2-(k-2)*x) + O(x*x^n))}
seq(n)= {concat([1], sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ))} \\ Andrew Howroyd, Feb 04 2022

a(n) = 2*A350354(n) for n >= 2. - Andrew Howroyd, Feb 04 2022

a(10)-a(18) from Alois P. Heinz, Dec 10 2021

Terms a(19) and beyond from Andrew Howroyd, Feb 04 2022

A335515 Number of patterns of length n matching the pattern (1,2,3).

Original entry on oeis.org

0, 0, 0, 1, 19, 257, 3167, 38909, 498235, 6811453, 100623211, 1612937661, 28033056683, 526501880989, 10639153638795, 230269650097469, 5315570416909995, 130370239796988957, 3385531348514480651, 92801566389186549245, 2677687663571344712043, 81124824154544921317597

Offset: 0

Gus Wiseman, Jun 19 2020

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(3) = 1 through a(4) = 19 patterns:
  (1,2,3)  (1,1,2,3)
           (1,2,1,3)
           (1,2,2,3)
           (1,2,3,1)
           (1,2,3,2)
           (1,2,3,3)
           (1,2,3,4)
           (1,2,4,3)
           (1,3,2,3)
           (1,3,2,4)
           (1,3,4,2)
           (1,4,2,3)
           (2,1,2,3)
           (2,1,3,4)
           (2,3,1,4)
           (2,3,4,1)
           (3,1,2,3)
           (3,1,2,4)
           (4,1,2,3)

Andrew Howroyd, Table of n, a(n) for n = 0..200
Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The complement A226316 is the avoiding version.
Compositions matching this pattern are counted by A335514 and ranked by A335479.
Permutations of prime indices matching this pattern are counted by A335520.
Patterns are counted by A000670 and ranked by A333217.
Patterns matching the pattern (1,1) are counted by A019472.
Permutations matching (1,2,3) are counted by A056986.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Cf. A034691, A158005, A158009, A238279, A333218, A333379, A333942.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

seq(n)=Vec( serlaplace(1/(2-exp(x + O(x*x^n)))) - 1/2 - 1/(1+sqrt(1-8*x+8*x^2 + O(x*x^n))), -(n+1)) \\ Andrew Howroyd, Jan 28 2024

a(n) = A000670(n) - A226316(n). - Andrew Howroyd, Jan 28 2024

a(9) onwards from Andrew Howroyd, Jan 28 2024

A335514 Number of (1,2,3)-matching compositions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 4, 14, 42, 114, 292, 714, 1686, 3871, 8696, 19178, 41667, 89386, 189739, 399144, 833290, 1728374, 3565148, 7319212, 14965880, 30496302, 61961380, 125577752, 253971555, 512716564, 1033496947, 2080572090, 4183940550, 8406047907, 16875834728

Offset: 0

Gus Wiseman, Jun 22 2020

nonn

			The a(6) = 1 through a(8) = 14 compositions:
  (1,2,3)  (1,2,4)    (1,2,5)
           (1,1,2,3)  (1,3,4)
           (1,2,1,3)  (1,1,2,4)
           (1,2,3,1)  (1,2,1,4)
                      (1,2,2,3)
                      (1,2,3,2)
                      (1,2,4,1)
                      (2,1,2,3)
                      (1,1,1,2,3)
                      (1,1,2,1,3)
                      (1,1,2,3,1)
                      (1,2,1,1,3)
                      (1,2,1,3,1)
                      (1,2,3,1,1)

Andrew Howroyd, Table of n, a(n) for n = 0..500
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

The version for permutations is A056986.
The avoiding version is A102726.
These compositions are ranked by A335479.
The version for patterns is A335515.
The version for prime indices is A335520.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Patterns matched by compositions are counted by A335456.
Cf. A011782, A032020, A106356, A226316, A269134, A333755, A335465, A335521.

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

a(n > 0) = 2^(n - 1) - A102726(n).

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020

A069321 Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)kk! for n >= 1.

Original entry on oeis.org

1, 1, 5, 31, 233, 2071, 21305, 249271, 3270713, 47580151, 760192505, 13234467511, 249383390393, 5057242311031, 109820924003705, 2542685745501751, 62527556173577273, 1627581948113854711, 44708026328035782905, 1292443104462527895991, 39223568601129844839353

Offset: 0

Karol A. Penson, Mar 14 2002

nonn

The number of compatible bipartitions of a set of cardinality n for which at least one subset is not underlined. E.g., for n=2 there are 5 such bipartitions: {1 2}, {1}{2}, {2}{1}, {1}{2}, {2}{1}. A005649 is the number of bipartitions of a set of cardinality n. A000670 is the number of bipartitions of a set of cardinality n with none of the subsets underlined. - Kyle Petersen, Mar 31 2005
a(n) is the cardinality of the image set summed over "all surjections". All surjections means: onto functions f:{1, 2, ..., n} -> {1, 2, ..., k} for every k, 1 <= k <= n.  a(n) = Sum_{k=1..n} A019538(n, k)*k. - Geoffrey Critzer, Nov 12 2012
From Gus Wiseman, Jan 15 2022: (Start)
For n > 1, also the number of finite sequences of length n + 1 covering an initial interval of positive integers with at least two adjacent equal parts, or non-anti-run patterns, ranked by the intersection of A348612 and A333217. The complement is counted by A005649. For example, the a(3) = 31 patterns, grouped by sum, are:
  (1111)  (1222)  (1122)  (1112)  (1233)  (1223)
          (2122)  (1221)  (1121)  (1332)  (1322)
          (2212)  (2112)  (1211)  (2133)  (2213)
          (2221)  (2211)  (2111)  (2331)  (2231)
          (1123)                  (3312)  (3122)
          (1132)                  (3321)  (3221)
          (2113)
          (2311)
          (3112)
          (3211)
Also the number of ordered set partitions of {1,...,n + 1} with two successive vertices together in some block.
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Benoit Cloitre, On the fractal behavior of primes, 2011. [internet archive]
Benoit Cloitre, On the fractal behavior of primes, 2011.
D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994.
D. Foata and D. Zeilberger, Graphical major indices, J. Comput. Appl. Math. 68 (1996), no. 1-2, 79-101.

Cf. A000629, A001563, A232472.
The complement is counted by A005649.
A version for permutations of prime indices is A336107.
A version for factorizations is A348616.
Dominated (n > 1) by A350252, complement A345194, compositions A345192.
A000670 = patterns, ranked by A333217.
A001250 = alternating permutations, complement A348615.
A003242 = anti-run compositions, ranked by A333489.
A019536 = necklace patterns.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A261983 = not-anti-run compositions, ranked by A348612.
A333381 = anti-runs of standard compositions.
Cf. A000110, A008277, A055932, A106356, A238424, A333382, A335456, A336103, A349050, A349055, A350138.

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> `if`(n=0, 2, b(n+1)-b(n))/2:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 02 2018

max = 20; t = Sum[n^(n - 1)x^n/n!, {n, 1, max}]; Range[0, max]!CoefficientList[Series[D[1/(1 - y(Exp[x] - 1)), y] /. y -> 1, {x, 0, max}], x] (* Geoffrey Critzer, Nov 12 2012 *)
Prepend[Table[Sum[StirlingS2[n, k]*k*k!, {k, n}], {n, 18}], 1] (* Michael De Vlieger, Jan 03 2016 *)
a[n_] := (PolyLog[-n-1, 1/2] - PolyLog[-n, 1/2])/4; a[0] = 1; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 30 2016 *)
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Length[Select[Join@@Permutations/@allnorm[n],MemberQ[Differences[#],0]&]],{n,0,8}] (* Gus Wiseman, Jan 15 2022 *)

{a(n)=polcoeff(1+sum(m=1, n, (2*m-1)!/(m-1)!*x^m/prod(k=1, m, 1+(m+k-1)*x+x*O(x^n))), n)} \\ Paul D. Hanna, Oct 28 2013

Representation as an infinite series: a(0) = 1 and a(n) = Sum_{k>=2} (k^n*(k-1)/(2^k))/4 for n >= 1. This is a Dobinski-type summation formula.
E.g.f.: (exp(x) - 1)/((2 - exp(x))^2).
a(n) = (1/2)*(A000670(n+1) - A000670(n)).
O.g.f.: 1 + Sum_{n >= 1} (2*n-1)!/(n-1)! * x^n / (Product_{k=1..n} (1 + (n + k - 1)*x)). - Paul D. Hanna, Oct 28 2013
a(n) = (A000629(n+1) - A000629(n))/4. - Benoit Cloitre, Oct 20 2002
a(n) = A232472(n-1)/2. - Vincenzo Librandi, Jan 03 2016
a(n) ~ n! * n / (4 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n > 0) = A000607(n + 1) - A005649(n). - Gus Wiseman, Jan 15 2022

A220097 Number of words on {1,1,2,2,3,3,...,n,n} avoiding the pattern 123.

Original entry on oeis.org

1, 1, 6, 43, 352, 3114, 29004, 280221, 2782476, 28221784, 291138856, 3045298326, 32222872906, 344293297768, 3709496350512, 40256666304723, 439645950112788, 4828214610825948, 53286643424088024, 590705976259292856, 6574347641664629388, 73433973722458186608

Offset: 0

Lara Pudwell, Dec 04 2012

nonn

a(n) is the number of 123-avoiding ordered set partitions of {1,...,2n} where all blocks are of size 2.

			For n=2, the a(2)=6 words are 1122, 1212, 1221, 2112, 2121, 2211.  For n=3, 213312 would be counted because it has no increasing subsequence of length 3, but 113223 would not be counted because it does have such an increasing subsequence.
For n=2, the a(2)=6 ordered set partitions are 12/34, 13/24, 14/23, 34/12, 24/13, 23/14.  For n=3, 46/23/15 would be counted because there is no way to choose i from the first block, j from the second block, and k from the third block such that i<j<k, but 13/25/46 would not be counted because we may select 1, 2, and 4 as a 123 pattern.

Alois P. Heinz, Table of n, a(n) for n = 0..931 (terms n=1..25 from Lara Pudwell)
Ferenc Balogh, A generalization of Gessel's generating function to enumerate words with double or triple occurrences in each letter and without increasing subsequences of a given length, preprint arXiv:1505.01389 [math.CO], 2015.
W. Y. C. Chen, A. Y. L. Dai and R. D. P. Zhou, Ordered Partitions Avoiding a Permutation of Length 3, arXiv preprint arXiv:1304.3187 [math.CO], 2013.
Anant Godbole, Adam Goyt, Jennifer Herdan, and Lara Pudwell, Pattern Avoidance in Ordered Set Partitions, arXiv preprint arXiv:1212.2530 [math.CO], 2012.
Robert A. Proctor, Matthew J. Willis, Convexity of tableau sets for type A Demazure characters (key polynomials), parabolic Catalan numbers, arXiv preprint arXiv:1706.03094 [math.CO], 2017.
Robert A. Proctor, Matthew J. Willis, Parabolic Catalan numbers count flagged Schur functions and their appearances as type A Demazure characters (key polynomials), arXiv:1706.04649 [math.CO], 2017.
Lara Pudwell, Enumeration schemes for words avoiding permutations, in Permutation Patterns (2010), S. Linton, N. Ruskuc, and V. Vatter, Eds., vol. 376 of London Mathematical Society Lecture Note Series, Cambridge University Press, pp. 193-211. Cambridge: Cambridge University Press.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
N. Shar, D. Zeilberger, The (Ordinary) Generating Functions Enumerating 123-Avoiding Words with r Occurrences of Each of 1, 2,..., n are Always Algebraic, arXiv preprint arXiv:1411.5052 [math.CO], 2014.

Cf. A266734, A266735, A226316.
Column k=2 of A267479.
Row sums of A288558.

Rest@ CoefficientList[Series[Sqrt[2/(1 + 2 x + Sqrt[1 - 12 x])], {x, 0, 20}], x] (* Michael De Vlieger, Oct 05 2016 *)
Table[Sum[(-1)^(n+k) Binomial[n,k]CatalanNumber[n+k], {k,0,n}], {n,1,20}] (* Peter Luschny, Aug 15 2017 *)

a(n) ~ 12^n/(sqrt(Pi)*(7*n/3)^(3/2)). - Vaclav Kotesovec, May 22 2013
G.f. = sqrt( 2/(1+2*x+sqrt(1-12*x))) [Chen et al.] - N. J. A. Sloane, Jun 09 2013
Conjecture: a(n) = (2/Pi)*Integral_{t=0..1} sqrt((1 - t)/t)*(16*t^2 - 4*t)^n = Catalan(2*n)*2F1(-1-2*n,-n;1/2-2*n;1/4). - Benedict W. J. Irwin, Oct 05 2016
a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(n,k)*Catalan(n+k). - Peter Luschny, Aug 15 2017
D-finite with recurrence: 4*n*(2*n+1)*a(n) +2*(-53*n^2+63*n-16)*a(n-1) +9*(13*n^2-59*n+62)*a(n-2) +18*(n-2)*(2*n-5)*a(n-3)=0. - R. J. Mathar, Feb 21 2020

a(0)=1 prepended by Alois P. Heinz, Nov 15 2019

A350252 Number of non-alternating patterns of length n.

Original entry on oeis.org

0, 0, 1, 7, 53, 439, 4121, 43675, 519249, 6867463, 100228877, 1602238783, 27866817297, 524175098299, 10606844137009, 229807953097903, 5308671596791901, 130261745042452855, 3383732450013895721, 92770140175473602755, 2677110186541556215233

Offset: 0

Gus Wiseman, Jan 13 2022

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217.
A sequence is alternating if it is alternately strictly increasing and strictly decreasing, starting with either. For example, the partition (3,2,2,2,1) has no alternating permutations, even though it does have the anti-run permutations (2,3,2,1,2) and (2,1,2,3,2). An alternating pattern is necessarily an anti-run (A005649).
Conjecture: Also the number of non-weakly up/down (or down/up) patterns of length n. For example:
- The a(3) = 7 non-weakly up/down patterns:
  (121), (122), (123), (132), (221), (231), (321)
- The a(3) = 7 non-weakly down/up patterns:
  (112), (123), (211), (212), (213), (312), (321)
- The a(3) = 7 non-alternating patterns (see example for more):
  (111), (112), (122), (123), (211), (221), (321)

			The a(2) = 1 and a(3) = 7 non-alternating patterns:
  (1,1)  (1,1,1)
         (1,1,2)
         (1,2,2)
         (1,2,3)
         (2,1,1)
         (2,2,1)
         (3,2,1)
The a(4) = 53 non-alternating patterns:
  2112   3124   4123   1112   2134   1234   3112   2113   1123
  2211   3214   4213   1211   2314   1243   3123   2123   1213
  2212   3412   4312   1212   2341   1324   3211   2213   1223
         3421   4321   1221   2413   1342   3212   2311   1231
                       1222   2431   1423   3213   2312   1232
                                     1432   3312   2313   1233
                                            3321   2321   1312
                                                   2331   1321
                                                          1322
                                                          1323
                                                          1332

Andrew Howroyd, Table of n, a(n) for n = 0..200

The unordered version is A122746.
The version for compositions is A345192, ranked by A345168, weak A349053.
The complement is counted by A345194, weak A349058.
The version for factorizations is A348613, complement A348610, weak A350139.
The strict case (permutations) is A348615, complement A001250.
The weak version for partitions is A349061, complement A349060.
The weak version for perms of prime indices is A349797, complement A349056.
The weak version is A350138.
The version for perms of prime indices is A350251, complement A345164.
A000670 = patterns (ranked by A333217).
A003242 = anti-run compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A019536 = necklace patterns.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
A226316 = patterns avoiding (1,2,3), weakly A052709, complement A335515.
A345163 = normal partitions w/ alternating permutation, complement A345162.
A345170 = partitions w/ alternating permutation, complement A345165.
A349055 = normal multisets w/ alternating permutation, complement A349050.
Cf. A049774, A052955, A096441, A128761, A274230, A335456, A335457, A336103, A344605, A344614.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
wigQ[y_]:=Or[Length[y]==0,Length[Split[y]]==Length[y]&& Length[Split[Sign[Differences[y]]]]==Length[y]-1];
Table[Length[Select[Join@@Permutations/@allnorm[n],!wigQ[#]&]],{n,0,6}]

a(n) = A000670(n) - A345194(n).

Terms a(9) and beyond from Andrew Howroyd, Feb 04 2022

cp's OEIS Frontend

A102726 Number of compositions of the integer n into positive parts that avoid a fixed pattern of three letters.

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A131689 Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

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A052709 Expansion of g.f. (1-sqrt(1-4*x-4*x^2))/(2*(1+x)).

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A344605 Number of alternating patterns of length n, including pairs (x,x).

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A345194 Number of alternating patterns of length n.

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A335515 Number of patterns of length n matching the pattern (1,2,3).

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A131689 Triangle of numbers T(n,k) = k!Stirling2(n,k) = A000142(k)A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.

A052709 Expansion of g.f. (1-sqrt(1-4x-4x^2))/(2*(1+x)).

A069321 Stirling transform of A001563: a(0) = 1 and a(n) = Sum_{k=1..n} Stirling2(n,k)kk! for n >= 1.