A005649 -id:A005649 - cp's OEIS frontend

A350138 Number of non-weakly alternating patterns of length n.

0, 0, 0, 2, 32, 338, 3560, 40058, 492664, 6647666, 98210192, 1581844994, 27642067000, 521491848218, 10572345303576, 229332715217954, 5301688511602448, 130152723055769810, 3381930236770946120, 92738693031618794378, 2676532576838728227352

Offset: 0

Gus Wiseman, Dec 24 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.
We define a sequence to be weakly alternating if it is alternately weakly increasing and weakly decreasing, starting with either.
Conjecture: The directed cases, which count non-weakly up/down or non-weakly down/up patterns, are both equal to the strong case: A350252.

			The a(4) = 32 patterns:
  (1,1,2,3)  (2,1,1,2)  (3,1,1,2)  (4,1,2,3)
  (1,2,2,1)  (2,1,1,3)  (3,1,2,3)  (4,2,1,3)
  (1,2,3,1)  (2,1,2,3)  (3,1,2,4)  (4,3,1,2)
  (1,2,3,2)  (2,1,3,4)  (3,2,1,1)  (4,3,2,1)
  (1,2,3,3)  (2,3,2,1)  (3,2,1,2)
  (1,2,3,4)  (2,3,3,1)  (3,2,1,3)
  (1,2,4,3)  (2,3,4,1)  (3,2,1,4)
  (1,3,2,1)  (2,4,3,1)  (3,3,2,1)
  (1,3,3,2)             (3,4,2,1)
  (1,3,4,2)
  (1,4,3,2)

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

The unordered version is A274230, complement A052955.
The strong case of compositions is A345192, ranked by A345168.
The strict case is A348615, complement A001250.
For compositions we have A349053, complement A349052, ranked by A349057.
The complement is counted by A349058.
The version for partitions is A349061, complement A349060.
The version for permutations of prime indices: A349797, complement A349056.
The version for ordered factorizations is A350139, complement A349059.
The strong case is A350252, complement A345194. Also the directed case?
A003242 = Carlitz compositions, complement A261983, ranked by A333489.
A005649 = anti-run patterns, complement A069321.
A025047/A129852/A129853 = alternating compositions, ranked by A345167.
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, A096441, A336103, A344605, A344614, A344615, A344740, A348610, A349794.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
whkQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]<=y[[m+1]],y[[m]]>=y[[m+1]]],{m,1,Length[y]-1}];
Table[Length[Select[Join@@Permutations/@allnorm[n],!whkQ[#]&&!whkQ[-#]&]],{n,0,6}]

R(n,k)={my(v=vector(k,i,1), u=vector(n)); for(r=1, n, if(r%2==0, my(s=v[k]); forstep(i=k, 2, -1, v[i] = s - v[i-1]); v[1] = s); for(i=2, k, v[i] += v[i-1]); u[r]=v[k]); u}
seq(n)= {concat([0], vector(n,i,1) + sum(k=1, n, (vector(n,i,k^i) - 2*R(n, k))*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) ) )} \\ Andrew Howroyd, Jan 13 2024

a(n) = A000670(n) - A349058(n).

a(9) onwards from Andrew Howroyd, Jan 13 2024

A350354 Number of up/down (or down/up) patterns of length n.

Original entry on oeis.org

1, 1, 1, 3, 11, 51, 281, 1809, 13293, 109899, 1009343, 10196895, 112375149, 1341625041, 17249416717, 237618939975, 3491542594727, 54510993341523, 901106621474801, 15723571927404189, 288804851413993941, 5569918636750820751, 112537773142244706427

Offset: 0

Gus Wiseman, Jan 16 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 patten is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase.
A pattern is up/down if it is alternately strictly increasing and strictly decreasing, starting with an increase. For example, the partition (3,2,2,2,1) has no up/down permutations, even though it does have the anti-run permutation (2,3,2,1,2).
Conjecture: Also the half the number of weakly up/down patterns of length n.
These are the values of the Euler zig-zag polynomials A205497 evaluated at x = 1/2 and normalized by 2^n. - Peter Luschny, Jun 03 2024

			The a(0) = 1 through a(4) = 11 patterns:
  ()  (1)  (1,2)  (1,2,1)  (1,2,1,2)
                  (1,3,2)  (1,2,1,3)
                  (2,3,1)  (1,3,1,2)
                           (1,3,2,3)
                           (1,3,2,4)
                           (1,4,2,3)
                           (2,3,1,2)
                           (2,3,1,3)
                           (2,3,1,4)
                           (2,4,1,3)
                           (3,4,1,2)

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

The version for permutations is A000111, undirected A001250.
For compositions we have A025048, down/up A025049, undirected A025047.
This is the up/down (or down/up) case of A345194.
A205497 are the Euler zig-zag polynomials.
A000670 counts patterns, ranked by A333217.
A005649 counts anti-run patterns.
A019536 counts necklace patterns.
A226316 counts patterns avoiding (1,2,3), weakly A052709.
A335515 counts patterns matching (1,2,3).
A349058 counts weakly alternating patterns.
A350252 counts non-alternating patterns.
Row sums of A079502.
Cf. A000041, A003242, A049774, A124754, A128761, A335457, A344604, A344605, A344615, A348610.

# Using the recurrence by Kyle Petersen from A205497.
G := proc(n) option remember; local F;
if n = 0 then 1/(1 - q*x) else F := G(n - 1);
simplify((p/(p - q))*(subs({p = q, q = p}, F) - subs(p = q, F))) fi end:
A350354 := n -> 2^n*subs({p = 1, q = 1, x = 1/2}, G(n)*(1 - x)^(n + 1)):
seq(A350354(n), n = 0..22);  # Peter Luschny, Jun 03 2024

allnorm[n_]:=If[n<=0,{{}},Function[s, Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
updoQ[y_]:=And@@Table[If[EvenQ[m],y[[m]]>y[[m+1]],y[[m]]

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, 0, F(k-2,-x)/F(k-1,x)-1) + 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) = A344605(n)/2.

a(n > 1) = A345194(n)/2.

Terms a(10) and beyond from Andrew Howroyd, Feb 04 2022

A083410 a(n) = A083385(n)/n.

Original entry on oeis.org

1, 4, 22, 154, 1306, 12994, 148282, 1908274, 27333706, 431220034, 7428550042, 138737478994, 2792050329706, 60231133487074, 1386484468239802, 33921605427779314, 878976357571495306, 24046780495646314114, 692622345890928153562, 20950628198687114521234, 663992311200423614606506

Offset: 1

N. J. A. Sloane, Jun 08 2003

nonn

From Michael Somos, Mar 04 2004: (Start)
Stirling transform of A052849(n+1)=[4,12,48,240,...] is 4*a(n)=[4,16,88,616,...].
Stirling transform of A001710(n+1)=[1,3,12,160,...] is a(n)=[1,4,22,154,...].
Stirling transform of A001563(n+1)=[4,18,96,600,...] is a(n+1)=[4,22,154,...]. (End)

N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.

A005649(n)=2*a(n), if n>0.
Pairwise sums of A091346.
Cf. A090665.

b:= proc(n, m) option remember;
     `if`(n=0, (m+1)!, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0)/2:
seq(a(n), n=1..23);  # Alois P. Heinz, Feb 14 2025

a[n_] := (-1)^n (PolyLog[-n - 1, 2] - PolyLog[-n, 2])/8;
Array[a, 21] (* Jean-François Alcover, Sep 10 2018, from A005649 *)

a(n)=if(n<0,0,n!*polcoeff(subst((1/(1-y)^2-1)/2,y,exp(x+x*O(x^n))-1),n))

E.g.f.: (1/(2-exp(x))^2-1)/2. - Michael Somos, Mar 04 2004
G.f.: 1/Q(0), where Q(k) = 1 - x*(3*k+4) - 2*x^2*(k+1)*(k+3)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 03 2013
a(n) ~ n! * n / (8 * (log(2))^(n+2)). - Vaclav Kotesovec, Jul 01 2018
a(n) = Sum_{k=1..n} k * A090665(n,k). - Alois P. Heinz, Feb 20 2025

A226738 Row 3 of array in A226513.

Original entry on oeis.org

1, 4, 24, 184, 1704, 18424, 227304, 3147064, 48278184, 812387704, 14872295784, 294192418744, 6251984167464, 142032703137784, 3434617880825064, 88075274293319224, 2387099326339205544, 68177508876215724664, 2046501717592969431144, 64408432189100396344504

Offset: 0

Vincenzo Librandi, Jun 18 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..100
Connor Ahlbach, Jeremy Usatine and Nicholas Pippenger, Barred Preferential Arrangements, Electron. J. Combin., Volume 20, Issue 2 (2013), #P55.

Cf. rows 0, 1, 2, 4, 5 of A226513: A000670, A005649, A226515, A226739, A226740.

m:=3; [&+[StirlingSecond(n,i)*Factorial(i)*Binomial(m+i,i): i in [0..n]]: n in [0..20]]; // Bruno Berselli, Jun 18 2013

Range[0, 20]! CoefficientList[Series[(2 - Exp@x)^-4, {x, 0, 20}], x]

E.g.f.: 1/(2 - exp(x))^4 (see the Ahlbach et al. paper, Theorem 4).
a(n) = sum( S2(n,i)*i!*binomial(3+i,i), i=0..n ), where S2 is the Stirling number of the second kind (see the Ahlbach et al. paper, Theorem 3). [Bruno Berselli, Jun 18 2013]
G.f.: 1/T(0), where T(k) = 1 - x*(k+4)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 28 2013
a(n) ~ n! * n^3 / (96 * log(2)^(n+4)). - Vaclav Kotesovec, Oct 11 2022
Conjectural g.f. as a continued fraction of Stieltjes type: 1/(1 - 4*x/(1 - 2*x/(1 - 5*x/(1 - 4*x/(1 - 6*x/(1 - 6*x/(1 - (n+3)*x/(1 - 2*n*x/(1 - ... ))))))))). - Peter Bala, Aug 27 2023
From Seiichi Manyama, Nov 19 2023: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} (3*k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) - 2*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). (End)

A217389 Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.

Original entry on oeis.org

1, 2, 5, 18, 93, 634, 5317, 52610, 598445, 7685706, 109933269, 1732565842, 29824133437, 556682481818, 11198025452261, 241481216430114, 5557135898411469, 135927902927547370, 3521462566184392693, 96323049885512803826, 2774010846129897006941, 83898835844633970888762

Offset: 0

Emanuele Munarini, Oct 02 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000670, A006957, A005649, A217388, A217391, A217392.

See A239914 for another version.

A000670:=func;
[&+[A000670(k): k in [0..n]]: n in [0..19]]; // Bruno Berselli, Oct 03 2012

b:= proc(n, k) option remember;
     `if`(n=0, k!, k*b(n-1, k)+b(n-1, k+1))
    end:
a:= proc(n) option remember; `if`(n<0, 0, a(n-1)+b(n, 0)) end:
seq(a(n), n=0..23);  # Alois P. Heinz, Feb 20 2025

t[n_] := Sum[StirlingS2[n, k]k!, {k, 0, n}]; Table[Sum[t[k], {k, 0, n}], {n, 0, 100}]
(* second program: *)
Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; Table[Fubini[n, 1], {n, 0, 20}] // Accumulate (* Jean-François Alcover, Mar 31 2016 *)

t(n):=sum(stirling2(n,k)*k!,k,0,n);
makelist(sum(t(k),k,0,n),n,0,40);

for(n=0,30, print1(sum(k=0,n, sum(j=0,k, j!*stirling(k,j,2))), ", ")) \\ G. C. Greubel, Feb 07 2018

a(n) = Sum_{k=0..n} t(k), where t = A000670 (ordered Bell numbers).
G.f. = A(x)/(1-x), where A(x) = g.f. for A000670 (see that entry). - N. J. A. Sloane, Apr 12 2014
a(n) ~ n! / (2* (log(2))^(n+1)). - Vaclav Kotesovec, Nov 08 2014

A337564 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal runs.

Original entry on oeis.org

1, 1, 6, 80, 1540, 38808, 1206744, 44595408, 1908389340, 92780281880, 5050066185736, 304196411024688, 20087958167374552, 1442953024024996400, 112007566256683719600, 9342904053303870936480, 833388624898522799682780, 79159669418651567937733080

Offset: 0

Gus Wiseman, Sep 03 2020

nonn

Sequences covering an initial interval of positive integers are counted by A000670 and ranked by A333217.

			The a(0) = 1 through a(2) = 6 sequences:
  ()  (1,1)  (1,1,1,2)
             (1,1,2,2)
             (1,2,2,2)
             (2,1,1,1)
             (2,2,1,1)
             (2,2,2,1)
The a(3) = 80 sequences:
  212222  111121  122233  333112  211133
  221222  111211  133222  333211  233111
  222122  112111  222133  112233  331112
  222212  121111  222331  113322  332111
  122221  123333  331222  221133  111223
  211222  133332  332221  223311  111322
  221122  213333  122223  331122  221113
  222112  233331  132222  332211  223111
  112221  333312  222213  112223  311122
  122211  333321  222231  113222  322111
  211122  122333  312222  222113  111123
  221112  133322  322221  222311  111132
  111221  221333  112333  311222  211113
  112211  223331  113332  322211  231111
  122111  333122  211333  111233  311112
  211112  333221  233311  111332  321111

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

A335461 has this as main diagonal n = 2*k.
A336108 is the version for compositions.
A337504 is the version for compositions and anti-runs.
A337505 is the version for anti-runs.
A000670 counts sequences covering an initial interval.
A005649 counts anti-runs covering an initial interval.
A124767 counts maximal runs in standard compositions.
A333769 gives run lengths in standard compositions.
A337504 counts compositions of 2*n with n maximal anti-runs.
A337565 gives anti-run lengths in standard compositions.
Cf. A001700, A003242, A052841, A060223, A106351, A106356, A269134, A325535, A333489, A333627, A333755, A335838.

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[2*n],Length[Split[#]]==n&]],{n,0,3}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
a(n) = {if(n==0, 1, b(n-1)*binomial(2*n-1,n-1))} \\ Andrew Howroyd, Dec 31 2020

a(n) = A005649(n-1)*binomial(2*n-1,n-1) = A005649(n-1)*A001700(n-1) for n > 0. - Andrew Howroyd, Dec 31 2020

Terms a(5) and beyond from Andrew Howroyd, Dec 31 2020

A348382 Number of compositions of n that are not a twin (x,x) but have adjacent equal parts.

Original entry on oeis.org

0, 0, 0, 1, 3, 9, 17, 41, 88, 185, 387, 810, 1669, 3435, 7039, 14360, 29225, 59347, 120228, 243166, 491085, 990446, 1995409, 4016259, 8076959, 16231746, 32599773, 65437945, 131293191, 263316897, 527912139, 1058061751, 2120039884, 4246934012, 8505864639

Offset: 0

Gus Wiseman, Nov 05 2021

nonn

A composition with no adjacent equal parts is also called a Carlitz composition, so these are non-twin, non-Carlitz compositions.

			The a(3) = 1 through a(6) = 17 compositions:
  (111)  (112)   (113)    (114)
         (211)   (122)    (222)
         (1111)  (221)    (411)
                 (311)    (1113)
                 (1112)   (1122)
                 (1121)   (1131)
                 (1211)   (1221)
                 (2111)   (1311)
                 (11111)  (2112)
                          (2211)
                          (3111)
                          (11112)
                          (11121)
                          (11211)
                          (12111)
                          (21111)
                          (111111)

A. Knopfmacher and H. Prodinger, On Carlitz Compositions, Europ. J. Combinatorics (1998) 19, 579-589.
Wikipedia, Composition (combinatorics)

Allowing twins gives A261983, complement A003242.
The non-alternating case is A348377, difference A345195.
These compositions are ranked by A348612 \ A007582.
A001250 counts alternating permutations, complement A348615.
A007582 ranks twin compositions.
A011782 counts compositions, strict A032020.
A025047 counts alternating or wiggly compositions, complement A345192.
A051049 counts non-twin compositions, complement A000035(n+1).
A325534 counts separable partitions, ranked by A335433.
A325535 counts inseparable partitions, ranked by A335448.
Cf. A000070, A005649, A059841, A106356, A238279, A333755, A344604, A344614, A344740, A348381.

nn=15;CoefficientList[Series[1+x/(1-2x)-x^2/(1-x^2)-1/(1-Sum[x^k/(1+x^k),{k,1,nn}]),{x,0,nn}],x]

For n > 0, a(n) = A261983(n) - A059841(n).

O.g.f.: 1 + x/(1-2x) - x^2/(1-x^2) - 1/(1 - Sum_{k>0} x^k/(1+x^k)).

A367470 Expansion of e.g.f. 1 / (3 - 2 * exp(x))^2.

Original entry on oeis.org

1, 4, 28, 268, 3244, 47404, 810988, 15891628, 350851564, 8615761324, 232911898348, 6872755977388, 219799913877484, 7572909749244844, 279630706025296108, 11016315458773541548, 461211305514352065004, 20448268640012928321964

Offset: 0

Seiichi Manyama, Nov 19 2023

nonn

Cf. A004123, A367471.
Cf. A005649, A367472.
Cf. A367474.

a(n) = sum(k=0, n, 2^k*(k+1)!*stirling(n, k, 2));

a(n) = Sum_{k=0..n} 2^k * (k+1)! * Stirling2(n,k).
a(0) = 1; a(n) = 2*Sum_{k=1..n} (k/n + 1) * binomial(n,k) * a(n-k).
a(0) = 1; a(n) = 4*a(n-1) - 3*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k).
a(n) ~ sqrt(2*Pi) * n^(n + 3/2) / (9 * log(3/2)^(n+2) * exp(n)). - Vaclav Kotesovec, May 20 2025

A095705 Triangular array read by rows: a(n, k) = sum of number of ordered factorizations of all prime signatures with n total prime factors and k distinct prime factors.

Original entry on oeis.org

1, 2, 3, 4, 8, 13, 8, 46, 44, 75, 16, 124, 308, 308, 541, 32, 572, 1790, 2536, 2612, 4683, 64, 1568, 8352, 17028, 24704, 25988, 47293, 128, 6728, 40628, 137498, 187928, 277456, 296564, 545835, 256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261

Offset: 1

Alford Arnold, Jul 04 2004, Nov 22 2005

A093936 is an analogous array for unordered factorizations.

First column is A000079. First two diagonals are A000670 and A005649.

			There are two prime signatures with 5 total primes and 3 distinct primes: p^3*q*r and p^2*q^2*r. A074206(p^3*q*r) = 132 and A074206(p^2*q^2*r) = 176, so a(5, 3) = 132+176 = 308.
Array begins:
  1
  2, 3
  4, 8, 13
  8, 46, 44, 75
  16, 124, 308, 308, 541
  32, 572, 1790, 2536, 2612, 4683
  64, 1568, 8352, 17028, 24704, 25988, 47293
  128, 6728, 40628, 137498, 187928, 277456, 296564, 545835
  256, 18768, 228308, 719056, 1699184, 2356560, 3526448, 3816548, 7087261

Amiram Eldar, Table of n, a(n) for n = 1..210 (rows n=1..20, flattened)

A035341 gives the row sums. Cf. A050324, A074206, A093936, A096443.

H[0] = 0; H[1] = 1; H[n_] := H[n] = Total[H /@ Most[Divisors[n]]];
T[n_, k_] := Module[{t = IntegerPartitions[n, {k}]}, Total[H /@ Times @@@ ((Prime[Range[k]]^#) & /@ t)]];
Table[T[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Amiram Eldar, Jun 28 2025 *)

Edited and extended by David Wasserman, Feb 22 2008

A337505 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal anti-runs.

Original entry on oeis.org

1, 2, 24, 440, 10780, 329112, 12006456, 508903824, 24559486380, 1328964785720, 79670488601704, 5240336913228144, 375167786246499064, 29038998659140223600, 2416268289647552828400, 215068032231876851531040, 20389611819955706893052460, 2051184695261785540782403320

Offset: 0

Gus Wiseman, Sep 05 2020

nonn

An anti-run is a sequence with no adjacent equal parts.

			The a(2) = 24 sequences:
  (2,1,2,2)  (1,2,3,3)  (1,2,2,3)  (1,1,2,3)
  (2,2,1,2)  (1,3,3,2)  (1,3,2,2)  (1,1,3,2)
  (1,2,2,1)  (2,1,3,3)  (2,2,1,3)  (2,1,1,3)
  (2,1,1,2)  (2,3,3,1)  (2,2,3,1)  (2,3,1,1)
  (1,1,2,1)  (3,3,1,2)  (3,1,2,2)  (3,1,1,2)
  (1,2,1,1)  (3,3,2,1)  (3,2,2,1)  (3,2,1,1)

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

A336108 is the version for compositions and runs.
A337504 is the version for compositions.
A337506 has this as main diagonal n = 2*k.
A337564 is the version for runs.
A000670 counts sequences covering an initial interval.
A003242 counts anti-run compositions.
A005649 counts anti-runs covering an initial interval.
A124767 counts maximal runs in standard compositions.
A333381 counts maximal anti-runs in standard compositions.
A333769 gives run-lengths in standard compositions.
A337565 gives anti-run lengths in standard compositions.
Cf. A052841, A106351, A106356, A269134, A325535, A333489, A333627, A333755.

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[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,3}]

\\ here b(n) is A005649.
b(n) = {sum(k=0, n, stirling(n,k,2)*(k + 1)!)}
a(n) = {b(n)*binomial(2*n-1,n)} \\ Andrew Howroyd, Dec 31 2020

a(n) = A005649(n)*binomial(2*n-1,n). - Andrew Howroyd, Dec 31 2020

Terms a(5) and beyond from Andrew Howroyd, Dec 31 2020

cp's OEIS Frontend

A350138 Number of non-weakly alternating patterns of length n.

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A350354 Number of up/down (or down/up) patterns of length n.

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A083410 a(n) = A083385(n)/n.

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A226738 Row 3 of array in A226513.

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A217389 Partial sums of the ordered Bell numbers (number of preferential arrangements) A000670.

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A337564 Number of sequences of length 2*n covering an initial interval of positive integers and splitting into n maximal runs.

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A348382 Number of compositions of n that are not a twin (x,x) but have adjacent equal parts.

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