A052709 -id:A052709 - cp's OEIS frontend

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.

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

A025227 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 3.

Original entry on oeis.org

0, 1, 2, 4, 12, 40, 144, 544, 2128, 8544, 35008, 145792, 615296, 2625792, 11311616, 49124352, 214838528, 945350144, 4182412288, 18593224704, 83015133184, 372090122240, 1673660915712, 7552262979584, 34178799378432, 155096251351040, 705533929816064

Offset: 0

Clark Kimberling

Series reversion of g.f. A(x) is -A(-x). - Michael Somos, Jul 27 2003
a(n) is the number of royal paths (A006318) from (0,0) to (n-1,n-1) such that every northeast (diagonal) step is either immediately followed by a north step or ends the path. For example a(3)=4 counts EDN, EENN, END, ENEN (E=east, D=diagonal, N=north). - David Callan, Jul 03 2006
From David Callan, Sep 25 2006: (Start)
a(n) is the number of ordered trees with n leaves in which (i) every node (= non-root non-leaf vertex) has at least 2 children and (ii) each leaf is either the leftmost or rightmost child of its parent. For example, a(3)=4 counts
              |
     /\      / \
      /\       /\
and their mirror images. (End)
From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010: (Start)
a(n+1), n >= 0, is also the number of Rota-Baxter words in one idempotent generator x and one operator of arity n.
Alternatively, a(n+1) is the number of ways of adding pairs of parentheses to a string of n x's (the number m of parentheses pairs necessarily satisfies m <= n <= 2m+1 for a nonzero count), such that no two pairs of parentheses are immediately nested and no two x's remain adjacent. (End)
a(n) is the number of colored binary trees on n-1 vertices where leaves have 2 possible colors and internal nodes have 1 color. - Alexander Burstein, Mar 07 2020

			For n=2, a(3) = 4 has the following words: x(x), (x)x, (x(x)), ((x)x) corresponding to A(1,2)=2, and A(2,2)=2. - William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010

L. Guo and W. Sit, Enumeration of Rota-Baxter Words (extended abstract), ISSAC 2006 Proceedings, 123-131. [From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010]
L. Guo and W. Sit, Enumeration of Rota-Baxter Words, to appear in Mathematics in Computer Science, Special Issue on AADIOS special session, ACA, 2009. [From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010]

Michael De Vlieger, Table of n, a(n) for n = 0..1470
Paul Barry, Riordan Pseudo-Involutions, Continued Fractions and Somos 4 Sequences, arXiv:1807.05794 [math.CO], 2018.
Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 22.
Maciej Dziemianczuk, On Directed Lattice Paths With Additional Vertical Steps, arXiv preprint arXiv:1410.5747 [math.CO], 2014.
Li Guo and William Y. Sit, Enumeration and generating functions of Rota-Baxter Words, Math. Comput. Sci. 4 (2010) 313-337.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 655
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 657
Donatella Merlini, Douglas G. Rogers, Renzo Sprugnoli, and M. Cecilia Verri, On some alternative characterizations of Riordan arrays, Canad. J. Math., 49 (1997), 301-320.

Cf. A052709, A052709, A100238.
Cf. A182399, A219534, A000108, A006318.
Cf. A068763.

Table[CatalanNumber[n-1] Hypergeometric2F1[(1-n)/2, -n/2, 3/2-n, -1] + KroneckerDelta[n], {n, 0, 20}] (* Vladimir Reshetnikov, May 17 2016 *)

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

a(n) = A052709(n) + A052709(n-1).
A100238(n) = -(-1)^n*a(n), for n>1.
a(n) = Sum_{k=0..floor(n/2)} C(n-k-1)*binomial(n-k, k), where C(q)=binomial(2q, q)/(q+1) are the Catalan numbers (A000108). - Emeric Deutsch, Nov 14 2001 [{a(n+1)}A068763.%20-%20_Wolfdieter%20Lang">{n>=0} = row sum of A068763. - _Wolfdieter Lang, Jan 21 2023]
D-finite with recurrence n*a(n) = (4n-6)*a(n-1)+(4n-12)*a(n-2), n>2. a(1)=1, a(2)=2.
G.f. satisfies A(x)-A(x)^2 = x+x^2. - Ralf Stephan, Jun 30 2003
a(n) = Sum_{k=0..n-1} C(k)*C(k+1, n-k-1). - Paul Barry, Feb 23 2005
G.f. A(x) satisfies A(x)=x+C(2x*A(x)) where C(x) is g.f. of Catalan numbers A000108 offset 1. - Michael Somos, Sep 08 2005
G.f.: (1-sqrt(1-4x-4x^2))/2 = 2(x+x^2)/(1+sqrt(1-4x-4x^2)). - Michael Somos, Jun 08 2000
Given an integer t >= 1 and initial values u = [a_0, a_1, ..., a_{t-1}], we may define an infinite sequence Phi(u) by setting a_n = a_{n-1} + a_0*a_{n-1} + a_1*a_{n-2} + ... + a_{n-2}*a_1 for n >= t. For example Phi([1]) is the Catalan numbers A000108. The present sequence is (essentially) Phi([1,2]). - Gary W. Adamson, Oct 27 2008
From William Sit (wyscc(AT)sci.ccny.cuny.edu), Jun 26 2010: (Start)
a(n+1), n >= 0, is column sum for the n-th column of the table R(m,n)=binomial(m+1, n-m)c(m) where c(m) is the m-th Catalan number A000108.
The table entry is nonzero if and only if m <= n <= 2m+1.
R(m,n) gives the number of Rota-Baxter words in one idempotent generator x and one operator of degree m and arity n, or the number of ways of adding m pairs of parentheses to a string of n x's (n necessarily lies between m and 2m+1 inclusive for a nonzero count), such that no two pairs of parentheses are immediately nested and no two x's remain adjacent. (End)
G.f.: A(x) = B(B(x)) where B(x) is the g.f. of A182399. -Paul D. Hanna, Apr 27 2012
G.f.: 1 - x + x*G(0), where G(k) = 1 + 1/(1 - (1+x)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 01 2013
a(n) ~ (1 + sqrt(2))^(n - 1/2) * 2^(n - 5/4) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 18 2013, simplified Jan 21 2023
O.g.f.: A(x) = x*S(x/(1 + x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318. - Peter Bala, Mar 05 2020
G.f.: A(x) satisfies ((A(x) - A(-x))/(2*x))^2 = S(4*x^2), where S(x) is the g.f. for the large Schröder numbers A006318. - Alexander Burstein, May 20 2021
A(x) = (x + x^2)*c(x+x^2), where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. Note that (x - x^2)*c(x-x^2) = x. - Peter Bala, Aug 29 2024

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

A073155 Leftmost column sequence of triangle A073153.

Original entry on oeis.org

1, 1, 4, 14, 56, 237, 1046, 4762, 22198, 105430, 508384, 2482297, 12248416, 60980875, 305955356, 1545397464, 7852100294, 40105277640, 205798130604, 1060467961508, 5485199090812, 28469067353686, 148220323891460

Offset: 0

Paul D. Hanna, Jul 29 2002

			a(3)=a(0)*[a(2)+a(1)]+[a(1)+a(0)]*[a(1)+a(0)]+[a(2)+a(1)]*a(0) =1*[4+1] + [1+1]*[1+1] + [4+1]*1 = 5 + 2*2 + 5 = 14.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
N. S. S. Gu, N. Y. Li and T. Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.

Cf. A073153, A073156, A073157.

Cf. A052709, A360076, A360082, A360083.

Convolution of sequence formed from sum of adjacent terms yields the original sequence without the first term:
a(n+1) = Sum_{k=0..n} [a(k) + a(k-1)] * [a(n-k) + a(n-k-1)], where a(-1)=0.
G.f.: 1/2*(1-(1-4*x*(1+x)^2)^(1/2))/x/(1+x)^2. - Vladeta Jovovic, Oct 10 2003
a(n) = Sum_{k=0..n} C(2k,n-k)*C(k). - Paul Barry, Jul 09 2006
Conjecture: (n+1)*a(n) + (-3*n+4)*a(n-1) + 2*(-6*n+7)*a(n-2) + 2*(-6*n+11)*a(n-3) + 2*(-2*n+5)*a(n-4)=0. - R. J. Mathar, Nov 26 2012
G.f. A(x) satisfies: A(x) = 1 + x * ((1 + x) * A(x))^2. - Ilya Gutkovskiy, Jul 10 2020

More terms from Vladeta Jovovic, Oct 10 2003

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

A366272 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^3A(x)^4.

Original entry on oeis.org

1, 1, 7, 49, 399, 3521, 32767, 316673, 3147775, 31977985, 330544639, 3465369601, 36759599103, 393820102657, 4255079743487, 46313023946753, 507319247208447, 5588706552643585, 61875364144283647, 688128167799619585, 7683686768042639359

Offset: 0

Seiichi Manyama, Oct 06 2023

nonn

Cf. A052709, A366221, A366273.

Cf. A366267, A366436.

a(n) = sum(k=0, n, binomial(3*k, n-k)*binomial(4*k, k)/(3*k+1));

a(n) = Sum_{k=0..n} binomial(3*k,n-k) * binomial(4*k,k)/(3*k+1).

G.f.: A(x) = 1/B(-x) where B(x) is the g.f. of A366436.

A071969 a(n) = Sum_{k=0..floor(n/3)} (binomial(n+1, k)binomial(2n-3k, n-3k)/(n+1)).

Original entry on oeis.org

1, 1, 2, 6, 19, 63, 219, 787, 2897, 10869, 41414, 159822, 623391, 2453727, 9733866, 38877318, 156206233, 630947421, 2560537092, 10435207116, 42689715279, 175243923783, 721649457417, 2980276087005, 12340456995177, 51222441676513, 213090270498764, 888321276659112

Offset: 0

N. J. A. Sloane, Jun 17 2002

nonn

Diagonal of A071946. - Emeric Deutsch, Dec 15 2004

Last (largest) number of each row of A071946. - David Scambler, May 15 2012

Seiichi Manyama, Table of n, a(n) for n = 0..1000
D. Merlini et al., Underdiagonal lattice paths with unrestricted steps, Discrete Appl. Math., 91 (1999), 197-213 (d_n page 209).

Cf. A071946 is the triangle and A119254 has the row sums.

Cf. A006318, A052709, A365268, A366025.

A071969 := n->add( binomial(n+1,k)*binomial(2*n-3*k,n-3*k)/(n+1),k=0..floor(n/3));
Order:=30: g:=solve(series((H-H^2)/(1+H^3),H)=z,H): seq(coeff(g,z^n),n=1..28); # Emeric Deutsch, Dec 15 2004

Table[Sum[Binomial[n+1,k] Binomial[2n-3k,n-3k]/(n+1),{k,0,Floor[n/3]}],{n,0,40}] (* Harvey P. Dale, Jul 20 2022 *)

a(n)=if(n<0,0,polcoeff(serreverse((x-x^2)/(1+x^3)+x^2*O(x^n)),n+1))

G.f. (offset 1) is series reversion of (x-x^2)/(1+x^3).

A085880 Triangle T(n,k) read by rows: multiply row n of Pascal's triangle (A007318) by the n-th Catalan number (A000108).

Original entry on oeis.org

1, 1, 1, 2, 4, 2, 5, 15, 15, 5, 14, 56, 84, 56, 14, 42, 210, 420, 420, 210, 42, 132, 792, 1980, 2640, 1980, 792, 132, 429, 3003, 9009, 15015, 15015, 9009, 3003, 429, 1430, 11440, 40040, 80080, 100100, 80080, 40040, 11440, 1430, 4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862

Offset: 0

N. J. A. Sloane, Aug 17 2003

Coefficients of terms in the series reversion of (1-k*x-(k+1)*x^2)/(1+x). - Paul Barry, May 21 2005
Equals A131427 * A007318 as infinite lower triangular matrices. [Philippe Deléham, Sep 15 2008]
Sum_{k=0..n} T(n,k)*x^k = A168491(n), A000007(n), A000108(n), A151374(n), A005159(n), A151403(n), A156058(n), A156128(n), A156266(n), A156270(n), A156273(n), A156275(n) for x = -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Nov 15 2013
Diagonal sums are A052709(n+1). - Philippe Deléham, Nov 15 2013

			Triangle starts:
[ 1]     1;
[ 2]     1,     1;
[ 3]     2,     4,      2;
[ 4]     5,    15,     15,      5;
[ 5]    14,    56,     84,     56,     14;
[ 6]    42,   210,    420,    420,    210,     42;
[ 7]   132,   792,   1980,   2640,   1980,    792,    132;
[ 8]   429,  3003,   9009,  15015,  15015,   9009,   3003,    429;
[ 9]  1430, 11440,  40040,  80080, 100100,  80080,  40040,  11440,  1430;
[10]  4862, 43758, 175032, 408408, 612612, 612612, 408408, 175032, 43758, 4862;
...

G. C. Greubel, Rows n = 0..100 of triangle, flattened
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.

Flat(List([0..10], n-> List([0..n], k-> Binomial(n,k)*Binomial(2*n,n)/( n+1) ))); # G. C. Greubel, Feb 07 2020

[Binomial(n,k)*Catalan(n): k in [0..n], n in [0..10]]; // G. C. Greubel, Feb 07 2020

seq(seq(binomial(n, k)*binomial(2*n, n)/(n+1), k = 0..n), n = 0..10); # G. C. Greubel, Feb 07 2020

Table[Binomial[n, k]*CatalanNumber[n], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Feb 07 2020 *)

tabl(nn) = {for (n=0, nn, c =  binomial(2*n,n)/(n+1); for (k=0, n, print1(c*binomial(n, k), ", ");); print(););} \\ Michel Marcus, Apr 09 2015

[[binomial(n,k)*catalan_number(n) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Feb 07 2020

Triangle given by [1, 1, 1, 1, 1, 1, ...] DELTA [1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
Sum_{k>=0} T(n, k) = A151374(n) (row sums). - Philippe Deléham, Aug 11 2005
G.f.: (1-sqrt(1-4*(x+y)))/(2*(x+y)). - Vladimir Kruchinin, Apr 09 2015

cp's OEIS Frontend

A073152 Triangle of numbers relating two simple context-free grammars (A052709 and A052705).

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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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A025227 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-1)*a(1) for n >= 3.

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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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A073155 Leftmost column sequence of triangle A073153.

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

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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.

A025227 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-1)*a(1) for n >= 3.

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

A366272 G.f. A(x) satisfies A(x) = 1 + x(1 + x)^3A(x)^4.

A071969 a(n) = Sum_{k=0..floor(n/3)} (binomial(n+1, k)binomial(2n-3k, n-3k)/(n+1)).