A001453 -id:A001453 - cp's OEIS frontend

A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

1, 2, 2, 3, 5, 5, 4, 9, 14, 14, 5, 14, 28, 42, 42, 6, 20, 48, 90, 132, 132, 7, 27, 75, 165, 297, 429, 429, 8, 35, 110, 275, 572, 1001, 1430, 1430, 9, 44, 154, 429, 1001, 2002, 3432, 4862, 4862, 10, 54, 208, 637, 1638, 3640, 7072, 11934, 16796, 16796, 11, 65, 273, 910

Offset: 1

Wolfdieter Lang, Jan 13 2006

First (k=0) column removed from Catalan triangle A009766(n,k).
In the Derrida et al. 1992 reference this triangle, called here X(alpha=1,beta=1;k=1,n,m), n >= m >= 1, is called there X_{N=n}(K=1,p=m) with alpha=1 and beta=1.
The column sequences give A000027 (natural numbers), A000096, A005586, A005587, A005557, A064059, A064061 for m=1..7. The numerator polynomials for the o.g.f. of column m is found in A062991 and the denominator is (1-x)^(m+1).
The diagonal sequences are convolutions of the Catalan numbers A000108, starting with the main diagonal.

			[1];[2,2];[3,5,5];[4,9,14,14];...
a(4,2) = 9 = binomial(6,2)*3/5.

B. Derrida, E. Domany and D. Mukamel, An exact solution of a one-dimensional asymmetric exclusion model with open boundaries, J. Stat. Phys. 69, 1992, 667-687; eqs. (20), (21), p. 672.
B. Derrida, M. R. Evans, V. Hakim and V. Pasquier, Exact solution of a 1D asymmetric exclusion model using a matrix formulation, J. Phys. A 26, 1993, 1493-1517; eq. (39), p. 1501, also appendix A1, (A12) p. 1513.

W. Lang: First 10 rows.

Row sums give A001453(n+1)=A000108(n+1)-1 (Catalan -1).

a(n, m)= binomial(n+m, n)*(n-m+1)/(n+1), n>=m>=1; a(n, m)=0 if n

A128634 Number of parallel permutations of length n.

Original entry on oeis.org

0, 2, 8, 26, 82, 262, 856, 2858, 9722, 33590, 117570, 416022, 1485798, 5348878, 19389688, 70715338, 259289578, 955277398, 3534526378, 13128240838, 48932534038, 182965127278, 686119227298, 2579808294646, 9723892802902, 36734706144302, 139067101832006, 527495903500718

Offset: 1

Ralf Stephan, May 08 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. Mansour and S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO], 2006.

Cf. A001453, A068875, A214292.

List([1..30], n-> 2*(Binomial(2*n, n)/(n+1) -1) ); # G. C. Greubel, Dec 02 2019

[2*(Catalan(n)-1): n in [1..40]]; // Vincenzo Librandi, Jul 22 2015

c:=binomial(2*n,n)/(n+1); seq(2*(c(n)-1), n=1..30); # G. C. Greubel, Dec 02 2019

Table[2 (CatalanNumber[n] - 1), {n, 30}] (* Vincenzo Librandi, Jul 22 2015 *)

vector(30, n, 2*(binomial(2*n,n)/(n+1) -1) ) \\ Michel Marcus, Jul 21 2015

[2*(catalan_number(n) -1) for n in (1..30)] # G. C. Greubel, Dec 02 2019

a(n) = -2 + 2 * binomial(2*n,n)/(n+1).
a(n) = -2 + A068875(n+1).
a(n) = 2*A001453(n) for n > 1. - J. M. Bergot, Sep 03 2013
a(n)= Sum_{r=0..n} A214292(n, r)^2. - J. M. Bergot, Sep 04 2013

More terms from Michel Marcus, Jul 21 2015

Offset changed by G. C. Greubel, Dec 02 2019

A141364 a(n) = C(n)-1+0^n where C(n) = A000108(n).

Original entry on oeis.org

1, 0, 1, 4, 13, 41, 131, 428, 1429, 4861, 16795, 58785, 208011, 742899, 2674439, 9694844, 35357669, 129644789, 477638699, 1767263189, 6564120419, 24466267019, 91482563639, 343059613649, 1289904147323, 4861946401451, 18367353072151

Offset: 0

Paul Barry, Jun 27 2008

Hankel transform is A141365.

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Cf. A000108, A001453 (essentially the same sequence), A141351.

CoefficientList[Series[(1 - Sqrt[1 - 4*x]) / (2*x)-x/(1-x),{x,0,26}],x] (* James C. McMahon, Jul 23 2025 *)

G.f.: c(x)-x/(1-x) where c(x) is the g.f. of A000108

A239432 Number of permutations of length n with longest increasing subsequence of length 8.

Original entry on oeis.org

1, 64, 2521, 79861, 2250887, 59367101, 1508071384, 37558353900, 927716186325, 22904111472825, 568209449266202, 14216730315766814, 359666061054003144, 9216708503647774264, 239524408949706575548, 6317740398995612513164, 169207499997274346326579, 4602911809939402715164066

Offset: 8

Vaclav Kotesovec, Mar 18 2014

nonn

In general, for column k of A047874 is a(n) ~ product(j!, j=0..k-1) * k^(2*n+k^2/2) / (2^((k-1)*(k+2)/2) * Pi^((k-1)/2) * n^((k^2-1)/2)) [Regev, 1981].

Vaclav Kotesovec, Table of n, a(n) for n = 8..135
A. Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), 115-136.

Cf. A047874, A001453, A001454, A001455, A001456, A001457, A001458.

a(n) ~ 1913625 * 2^(6*n+77) / (Pi^(7/2) * n^(63/2)).

A247493 Triangle read by rows: T(n, k) = C(n, k)C(2k, k)/(k+1) - sum(j = 0..k, (-1)^j(1-j)^nC(k, j)/k!), 0<=k<=n.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 2, 6, 4, 0, 3, 11, 22, 13, 0, 4, 20, 45, 75, 41, 0, 5, 29, 110, 190, 261, 131, 0, 6, 42, 154, 560, 826, 938, 428, 0, 7, 55, 322, 749, 2646, 3570, 3452, 1429, 0, 8, 72, 335, 2499, 3885, 12012, 15198, 12897, 4861, 0, 9, 89, 770, 650, 16947, 21693, 53880, 63915, 48655, 16795, 0, 10, 110, 484, 11660, -8338, 97482

Offset: 0

Peter Luschny, Oct 02 2014

First negative value appears at T(11,5). - Indranil Ghosh, Mar 04 2017

			0;
0, 0;
0, 1, 1;
0, 2, 6, 4;
0, 3, 11, 22, 13;
0, 4, 20, 45, 75, 41;
0, 5, 29, 110, 190, 261, 131;
0, 6, 42, 154, 560, 826, 938, 428;

Indranil Ghosh, Rows 0..100, flattened

Cf. A001453, A098474, A105794, A247491.

T := proc(n, k) binomial(n,k)*binomial(2*k,k)/(k+1) - add((-1)^j*(1-j)^n /(j!*(k-j)!), j = 0..k) end:
for n from 0 to 12 do seq(T(n,k), k=0..n) od;

Flatten[Table[(Binomial[n,k] * Binomial[2k,k] / (k+1)) - Sum[(-1)^j*(1-j)^n*Binomial[k,j]/k!,{j,0,k}],{n,0,10},{k,0,n}]] (* Indranil Ghosh, Mar 04 2017 *)

tabl(nn) = {for (n=0, nn, for (k=0, n, print1((binomial(n,k)*binomial(2*k,k)/(k+1))-sum(j=0, k, (-1)^j*(1-j)^n*binomial(k,j)/k!),", ",);); print(););};
tabl(10); \\ Indranil Ghosh, Mar 04 2017

A105794(n, k) = (-1)^(n-k)*(C(n, k)*Catalan(k) - T(n, k)).
A247491(n) = Sum(k=0..n, (-1)^(n-k+1)*T(n, k)).
A001453(n) = T(n, n).
T(n,k) = A098474 (n,k) - A105794 (n,k). - Michel Marcus, Mar 04 2017

A349740 Number of partitions of set [n] in a set of <= k noncrossing subsets. Number of Dyck n-paths with at most k peaks. Both with 0 <= k <= n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 4, 5, 0, 1, 7, 13, 14, 0, 1, 11, 31, 41, 42, 0, 1, 16, 66, 116, 131, 132, 0, 1, 22, 127, 302, 407, 428, 429, 0, 1, 29, 225, 715, 1205, 1401, 1429, 1430, 0, 1, 37, 373, 1549, 3313, 4489, 4825, 4861, 4862, 0, 1, 46, 586, 3106, 8398, 13690, 16210, 16750, 16795, 16796

Offset: 0

Ron L.J. van den Burg, Nov 28 2021

Given a partition P of the set {1,2,...,n}, a crossing in P are four integers [a, b, c, d] with 1 <= a < b < c < d <= n for which a, c are together in a block, and b, d are together in a different block. A noncrossing partition is a partition with no crossings.

			For n=4 the T(4,3)=13 partitions are {{1,2,3,4}}, {{1,2,3},{4}}, {{1,2,4},{3}}, {{1,3,4},{2}}, {{2,3,4},{1}}, {{1,2},{3,4}}, {{1,4},{2,3}}, {{1,2},{3},{4}}, {{1,3},{2},{4}}, {{1,4},{2},{3}}, {{1},{2,3},{4}}, {{1},{2,4},{3}}, {{1},{2},{3,4}}.
The set of sets {{1,3},{2,4}} is missing because it is crossing. If you add the set of 4 sets, {{1},{2},{3},{4}}, you get T(4, 4) = 14 = A000108(4), the 4th Catalan number.
Triangle begins:
  1;
  0, 1;
  0, 1,  2;
  0, 1,  4,   5;
  0, 1,  7,  13,   14;
  0, 1, 11,  31,   41,   42;
  0, 1, 16,  66,  116,  131,  132;
  0, 1, 22, 127,  302,  407,  428,  429;
  0, 1, 29, 225,  715, 1205, 1401, 1429, 1430;
  0, 1, 37, 373, 1549, 3313, 4489, 4825, 4861, 4862;
  ...

Alois P. Heinz, Rows n = 0..200, flattened
David Callan, Sets, Lists and Noncrossing Partitions, Journal of Integer Sequences, Vol. 11 (2008), Article 08.1.3; also on arXiv, arXiv:0711.4841 [math.CO], 2007-2008.

Columns k=0-4 give (for n>=k): A000007, A000012, A000124(n-1), A116701, A116844.
Partial sums of A090181 per row.
Main diagonal is A000108.
Row sums give A088218.
T(2*n,n) gives A065097.
T(n,n-1) gives A001453 for n >= 2.
Cf. A106396, A137940.

b:= proc(x, y, t) option remember; expand(`if`(y<0
      or y>x, 0, `if`(x=0, 1, add(b(x-1, y+j, j)*
     `if`(t=1 and j<1, z, 1), j=[-1, 1]))))
    end:
T:= proc(n, k) option remember; `if`(k<0, 0,
      T(n, k-1)+coeff(b(2*n, 0$2), z, k))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Nov 28 2021

T[n_, k_] := If[n == 0, 1, Sum[j Binomial[n, j]^2 / (n - j + 1), {j, 0, k}] / n];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Peter Luschny, Nov 29 2021 *)

T(n,k) = Sum_{j=0..k} A090181(n,j), the partial sum of the Narayana numbers.
T(n,n) = A000108(n), the n-th Catalan number.
G.f.: (1 + x - x*y - sqrt((1-x*(1+y))^2 - 4*y*x^2))/(2*x*(1-y)).
T(n,k) = (1/n)*Sum_{j=0..k} j*binomial(n,j)^2 / (n-j+1) for n >= 1. - Peter Luschny, Nov 29 2021

A097607 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having leftmost valley at altitude k (if path has no valleys, then this altitude is considered to be 0).

Original entry on oeis.org

1, 1, 2, 4, 1, 9, 4, 1, 23, 13, 5, 1, 65, 41, 19, 6, 1, 197, 131, 67, 26, 7, 1, 626, 428, 232, 101, 34, 8, 1, 2056, 1429, 804, 376, 144, 43, 9, 1, 6918, 4861, 2806, 1377, 573, 197, 53, 10, 1, 23714, 16795, 9878, 5017, 2211, 834, 261, 64, 11, 1, 82500, 58785, 35072

Offset: 0

Emeric Deutsch, Aug 30 2004

Row sums are the Catalan numbers (A000108) Column 0 is A014137 (partial sums of Catalan numbers). Column 1 is A001453 (Catalan numbers -1).

			Triangle starts:
1;
1;
2;
4,1;
9,4,1;
23,13,5,1;
65,41,19,6,1;
T(4,1)=4 because we have UU(DU)DDUD, UU(DU)DUDD, UU(DU)UDDD and UUUD(DU)DD, where U=(1,1), D=(1,-1); the first valleys, all at altitude 1, are shown between parentheses.

Cf. A000108, A014137, A001453.

G.f.=(1-z+zC-tzC)/[(1-z)(1-tzC)], where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

A112307 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n with height of second peak equal to k (n>=1; 0<=k<=n-1).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 6, 3, 1, 9, 16, 12, 4, 1, 23, 44, 39, 20, 5, 1, 65, 128, 123, 76, 30, 6, 1, 197, 392, 393, 268, 130, 42, 7, 1, 626, 1250, 1284, 928, 505, 204, 56, 8, 1, 2056, 4110, 4287, 3216, 1880, 864, 301, 72, 9, 1, 6918, 13834, 14583, 11224, 6885, 3438, 1379

Offset: 1

Emeric Deutsch, Nov 30 2005

Row sums are the Catalan numbers (A000108). T(n,0)=1 (paths have only one peak); The g.f. for column k is kz^(k+1)*c^k/(1-z), where c=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,1)=A014137(n-1); T(n,2)=2*A014138(n-3); T(n,3)=3*A001453(n-2); T(n,4)=4*A114277(n-5); Sum(k*T(n,k), k=0..n-1)=A112308(n-2).

			T(4,1)=4 because we have UDUDUDUD, UDUDUUDD, UUDDUDUD and UUUDDDUD, where U=(1,1), D=(1,-1).
Triangle begins:
1;
1,1;
1,2,2;
1,4,6,3;
1,9,16,12,4;

Cf. A000108, A014137, A014138, A001453, A114277, A112308.

G:=((1-t*z*c)^2+t*z^2*c)/(1-z)/(1-t*z*c)^2-1: c:=(1-sqrt(1-4*z))/2/z: Gser:=simplify(series(G,z=0,15)): for n from 1 to 12 do P[n]:=coeff(Gser,z^n) od: for n from 1 to 12 do seq(coeff(t*P[n],t^j),j=1..n) od; # yields sequence in triangular form

G.f.=[(1-tzc)^2+tz^2*c]/[(1-z)(1-tzc)^2]-1, where c=[1-sqrt(1-4z)]/(2z) is the Catalan function.

A114276 Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having length of second ascent equal to k (0<=k<=n-1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 4, 1, 1, 22, 13, 5, 1, 1, 64, 41, 19, 6, 1, 1, 196, 131, 67, 26, 7, 1, 1, 625, 428, 232, 101, 34, 8, 1, 1, 2055, 1429, 804, 376, 144, 43, 9, 1, 1, 6917, 4861, 2806, 1377, 573, 197, 53, 10, 1, 1, 23713, 16795, 9878, 5017, 2211, 834, 261, 64, 11, 1, 1

Offset: 1

Emeric Deutsch, Nov 20 2005

Column 1 yields A014138, column 2 yields A001453, column 3 yields A114277. Row sums are the Catalan numbers (A000108).

			T(4,2)=4 because we have UD(UU)DDUD, UD(UU)DUDD, UUD(UU)DDD and UUDD(UU)DD (second ascent shown between parentheses).

Cf. A000108, A014138, A001453, A114277.

T:=proc(n,k) if k=0 then 1 elif k<=n-1 then (k+1)*sum(binomial(2*n-k+1-2*j,n-j+1)/(2*n-k-2*j+1),j=1..n-k) else 0 fi end: for n from 1 to 12 do seq(T(n,k),k=0..n-1) od; # yields sequence in triangular form

T(n, k)=(k+1)*sum(binomial(2*n-k+1-2*j, n-j+1)/(2*n-k-2*j+1), j=1..n-k) if 1<=k<=n-1; T(n, 0)=1. G.f. = (1-tz)/[(1-z)(1-tzC)]-1 where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

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A064308 Product of two triangular matrices C*S2.

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A115126 First (k=1) triangle of numbers related to totally asymmetric exclusion process (case alpha=1, beta=1).

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A128634 Number of parallel permutations of length n.

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A141364 a(n) = C(n)-1+0^n where C(n) = A000108(n).

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A239432 Number of permutations of length n with longest increasing subsequence of length 8.

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A247493 Triangle read by rows: T(n, k) = C(n, k)*C(2*k, k)/(k+1) - sum(j = 0..k, (-1)^j*(1-j)^n*C(k, j)/k!), 0<=k<=n.

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A349740 Number of partitions of set [n] in a set of <= k noncrossing subsets. Number of Dyck n-paths with at most k peaks. Both with 0 <= k <= n, read by rows.

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A097607 Triangle read by rows: T(n,k) is number of Dyck paths of semilength n and having leftmost valley at altitude k (if path has no valleys, then this altitude is considered to be 0).

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A247493 Triangle read by rows: T(n, k) = C(n, k)C(2k, k)/(k+1) - sum(j = 0..k, (-1)^j(1-j)^nC(k, j)/k!), 0<=k<=n.