A003517 -id:A003517 - cp's OEIS frontend

A050144 T(n,k) = M(2n-1,n-1,k-1), 0 <= k <= n, n >= 0, where M(p,q,r) is the number of upright paths from (0,0) to (p,p-q) that meet the line y = x+r and do not rise above it.

0, 1, 0, 1, 1, 1, 2, 3, 4, 1, 5, 9, 14, 6, 1, 14, 28, 48, 27, 8, 1, 42, 90, 165, 110, 44, 10, 1, 132, 297, 572, 429, 208, 65, 12, 1, 429, 1001, 2002, 1638, 910, 350, 90, 14, 1, 1430, 3432, 7072, 6188, 3808, 1700, 544, 119, 16, 1

Offset: 0

Clark Kimberling

Let V=(e(1),...,e(n)) consist of q 1's and p-q 0's; let V(h)=(e(1),...,e(h)) and m(h)=(#1's in V(h))-(#0's in V(h)) for h=1,...,n. Then M(p,q,r)=number of V having r=max{m(h)}.
The interpretation of T(n,k) as RU walks in terms of M(.,.,.) in the NAME is erroneous. There seems to be a pattern along subdiagonals:
 M(3,1,1) = 4  = T(3,2); M(3,1,2) = 1  = T(4,4); M(5,2,1) = 20 = T(5,3); M(5,2,2) = 7  = T(6,5); M(5,2,3) = 1  = T(7,7); M(7,3,0) = 165 = T(6,2); M(7,3,1) = 110 = T(7,4); M(7,3,2) = 44  = T(8,6); M(7,3,3) = 10  = T(9,8); M(7,3,4) = 1   = T(10,10); M(9,4,0) = 1001 = T(8,3); M(9,4,1) = 637  = T(9,5); M(9,4,2) = 273  = T(10,7); M(9,4,3) = 77   = T(11,9); M(9,4,4) = 13   = T(12,11); M(9,4,5) = 1   = T(13,13); M(11,5,0) = 6188 = T(10,4); M(11,5,1) = 3808 = T(11,6); M(11,5,2) = 1700 = T(12,8); M(11,5,3) = 544  = T(13,...); M(11,5,4) = 119; M(11,5,5) = 16; M(11,5,6) = 1; M(13,6,0) = 38760 = T(12,5); M(13,6,1) = 23256 = T(13,7); M(13,6,2) = 10659 = T(14,9); - R. J. Mathar, Jul 31 2024

			Triangle begins:
     0
     1    0
     1    1    1
     2    3    4    1
     5    9   14    6    1
    14   28   48   27    8    1
    42   90  165  110   44   10    1
   132  297  572  429  208   65   12    1
   429 1001 2002 1638  910  350   90   14    1
  1430 3432 7072 6188 3808 1700  544  119   16    1

Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.

Emeric Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
Boris A. Bondarenko, Generalized Pascal Triangles and Pyramids, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 29.
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

{M(2n, 0, k)} is given by A039599. {M(2n+1, n+1, k+1)} is given by A039598.

Cf. A033184, A050153, A000108 (column 0), A000245 (column 1), A002057 (column 2), A000344 (column 3), A003517 (column 4), A000588 (column 5), A003518 (column 6), A001392 (column 7), A003519 (column 8), A000589 (column 9), A090749 (column 10).

A050144 := proc(n,k)
    if n < k then
        0;
    elif k =0 then
        if n =0 then
            0 ;
        else
            A000108(n-1) ;
        end if;
    elif k = 1 then
        add( procname(n-1-j,0)*A000108(j+1),j=0..n-1) ;
    elif k = 2 then
        add( procname(n-j,1)*A000108(j),j=0..n) ;
    else
        add( procname(n-1-j,k-1)*A000108(j),j=0..n-1) ;
    end if;
end proc:
seq(seq( A050144(n,k),k=0..n),n=0..12) ; # R. J. Mathar, Jul 30 2024

c[n_] := Binomial[2 n, n]/(n + 1);
t[n_, k_] := Which[k == 0, c[n - 1],
  k == 1, Sum[t[n - 1 - j, 0]*c[j + 1], {j, 0, n - 2}],
  k == 2, Sum[t[n - j, 1]*c[j], {j, 0, n - 1}],
  k > 2, Sum[t[n - 1 - j, k - 1] c[j + 1], {j, 0, n - 2}]]
t[0, 0] = 0;
Column[Table[t[n, k], {n, 0, 10}, {k, 0, n}]]
(* Clark Kimberling, Jul 30 2024 *)

For n > 0: Sum_{k>=0} T(n, k) = binomial(2*n-1, n); see A001700. - Philippe Deléham, Feb 13 2004 [Erroneous sum-formula deleted. R. J. Mathar, Jul 31 2024]
T(n, k)=0 if n < k; T(0, 0)=0, T(n, 0) = A000108(n-1) for n > 0; T(n, 1) = Sum_{j>=0} T(n-1-j, 0)*A000108(j+1); T(n, 2) = Sum_{j>=0} T(n-j, 1)*A000108(j); for k > 2, T(n, k) = Sum_{j>=0} T(n-1-j, k-1)*A000108(j+1). - Philippe Deléham, Feb 13 2004 [Corrected by Sean A. Irvine, Aug 08 2021]
For the column k=0, g.f.: x*C(x); for the column k=1, g.f.: x*C(x)*(C(x)-1); for the column k, k > 1, g.f.: x*C(x)^2*(C(x)-1)^(k-1); where C(x) = Sum_{n>=0} A000108(n)*x^n is g.f. for Catalan numbers, A000108. - Philippe Deléham, Feb 13 2004
T(n,0) = A033184(n,2). T(n,1) = A033184(n+1,3), T(n,k) = A033184(n+2,k+2) for k>=2. - R. J. Mathar, Jul 31 2024

A138159 Triangle read by rows: T(n,k) is the number of permutations of [n] having k occurrences of the pattern 321 (n>=1, 0<=k<=n(n-1)(n-2)/6).

Original entry on oeis.org

1, 1, 2, 5, 1, 14, 6, 3, 0, 1, 42, 27, 24, 7, 9, 6, 0, 4, 0, 0, 1, 132, 110, 133, 70, 74, 54, 37, 32, 24, 12, 16, 6, 6, 8, 0, 0, 5, 0, 0, 0, 1, 429, 429, 635, 461, 507, 395, 387, 320, 260, 232, 191, 162, 104, 130, 100, 24, 74, 62, 18, 32, 10, 30, 13, 8, 0, 10, 10, 0, 0, 0, 6, 0, 0, 0, 0, 1

Offset: 0

Emeric Deutsch, Mar 27 2008

Row n has 1 + n(n-1)(n-2)/6 terms.
Sum of row n is n! (A000142).
T(n,0) = A000108(n) (the Catalan numbers).
T(n,1) = A003517(n-1).
T(n,2) = A001089(n).
Sum_{k>=0} k * T(n,k) = A001810(n).

			T(4,2) = 3 because we have 4312, 4231 and 3421.
Triangle starts:
    1;
    1;
    2;
    5,   1;
   14,   6,   3,  0,  1;
   42,  27,  24,  7,  9,  6,  0,  4,  0,  0,  1;
  132, 110, 133, 70, 74, 54, 37, 32, 24, 12, 16, 6, 6, 8, 0, 0, 5, 0, 0, 0, 1;
  ...

Alois P. Heinz, Rows n = 0..15, flattened
D. Callan, A recursive bijective approach to counting permutations containing 3-letter patterns, arXiv:math/0211380 [math.CO], 2002.
FindStat - Combinatorial Statistic Finder, The number of occurrences of the pattern [1,2,3] inside a permutation of length at least 3
M. Fulmek, Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three, Adv. Appl. Math., 30, 2003, 607-632. also Arxiv CO/0112092.
Toufik Mansour, Sherry H. F. Yan and Laura L. M. Yang, Counting occurrences of 231 in an involution, Discrete Mathematics 306 (2006), pages 564-572.
J. Noonan, The number of permutations containing exactly one increasing subsequence of length three, Discrete Math. 152 (1996), no. 1-3, 307-313.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998.
J. Noonan and D. Zeilberger, The enumeration of permutations with a prescribed number of "forbidden" patterns, Adv. Appl. Math., 17, 1996, 381-407.

Cf. A000108, A000142, A000292, A003517, A001089, A001810, A056986, A263771.

# The following Maple program yields row 9 of the triangle; change the value of n to obtain other rows.
n:=9: with(combinat): P:=permute(n): f:=proc(k) local L: L:=proc(j) local ct, i: ct:=0: for i to j-1 do if P[k][j] < P[k][i] then ct:=ct+1 else end if end do: ct end proc: add(L(j)*(L(j)+P[k][j]-j),j=1..n) end proc: a:=sort([seq(f(k),k=1..factorial(n))]): for h from 0 to (1/6)*n*(n-1)*(n-2) do c[h]:=0: for m to factorial(n) do if a[m]=h then c[h]:=c[h]+1 else end if end do end do: seq(c[h],h=0..(1/6)*n*(n-1)*(n-2));
# second Maple program:
b:= proc(s, c) option remember; (n-> `if`(n=0, x^c, add(b(s minus {j},
      (t-> (j-n+t)*t+c)(nops(select(x-> x>j, s)))), j=s)))(nops(s))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b({$1..n}, 0)):
seq(T(n), n=0..9);  # Alois P. Heinz, Dec 01 2021

ro[n_] := With[{}, P = Permutations[Range[n]]; f[k_] := With[{}, L[j_] := With[{}, ct = 0; Do[If[P[[k, j]] < P[[k, i]], ct = ct + 1], {i, 1, j - 1}]; ct]; Sum[L[j]*(L[j] + P[[k, j]] - j), {j, 1, n}]]; a = Sort[Table[f[k], {k, 1, n!}]]; Do[c[h] = 0; Do[If[a[[m]] == h, c[h] = c[h] + 1], {m, 1, n!}], {h, 0, (1/6)*n*(n - 1)*(n - 2)}]; Table[c[h], {h, 0, (1/6)*n*(n - 1)*(n - 2)}]]; Flatten[Table[ro[n], {n, 1, 7}]] (* Jean-François Alcover, Sep 01 2011, after Maple *)

The number of 321-patterns of a given permutation p of [n] is given by Sum(L[i]R[i],i=1..n), where L (R) is the left (right) inversion vector of p. L and R are related by R[i]+i=p[i]+L[i] (the given Maple program makes use of this approach). References contain formulas and generating functions for the first few columns (some are only conjectured).

A052720 Expansion of e.g.f.: (1 - 4x + 3x^2)(1 - 2x - sqrt(1-4x))/2 - x^2 + 2x^3.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 720, 30240, 1088640, 39916800, 1556755200, 65383718400, 2964061900800, 144815595724800, 7602818775552000, 427447366714368000, 25646842002862080000, 1636734826000834560000, 110752389892723138560000

Offset: 0

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

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 676

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052721, A052722, A052723.

Cf. A000108, A003517.

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

Table[If[n<6, 0, 6*(n-2)!*Binomial[n-4, 2]*CatalanNumber[n-3]], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052720(n):
    if (n<6): return 0
    else: return 6*factorial(n-2)*binomial(n-4,2)*catalan_number(n-3)
[A052720(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(1)=0; a(2)=0; a(4)=0; a(3)=0; a(5)=0; a(6)=720; a(n+3) = (10+8*n)*a(n+2) + (22-27*n-19*n^2)*a(n+1) - (60-66*n+6*n^2+12*n^3)*a(n).
a(n) = n!*A003517(n-4). - R. J. Mathar, Oct 18 2013
From G. C. Greubel, May 28 2022: (Start)
G.f.: 6!*x^6*Hypergeometric2F0([3, 7/2], [], 4*x).
E.g.f.: (1/2)*(1 - 6*x + 9*x^2 - 2*x^3 - (1 - 4*x + 3*x^2)*sqrt(1-4*x)). (End)

A120009 G.f.: A(x) = (x-x^2) o x/(1-x) o (1-sqrt(1-4*x))/2, a composition of functions involving the Catalan function and its inverse.

Original entry on oeis.org

1, 1, 1, 0, -6, -33, -143, -572, -2210, -8398, -31654, -118864, -445740, -1671525, -6273135, -23571780, -88704330, -334347090, -1262330850, -4773905760, -18083762580, -68611922730, -260725306374, -992233959480, -3781513867796, -14431491699548, -55147299002348

Offset: 1

Paul D. Hanna, Jun 03 2006

The n-th self-composition of A(x) is: (x-x^2) o x/(1-n*x) o (1-sqrt(1-4*x))/2. See A120010 for the transpose of the composition of the same functions.

			A(x) = x + x^2 + x^3 - 6*x^5 - 33*x^6 - 143*x^7 - 572*x^8 - 2210*x^9 + ...
A(x) = x*C(x)^2 - x^2*C(x)^4 where C(x) is Catalan function so that:
x*C(x)^2 = x + 2*x^2 + 5*x^3 + 14*x^4 + 42*x^5 + 132*x^6 + 429*x^7 + ...
x^2*C(x)^4 = x^2 + 4*x^3 + 14*x^4 + 48*x^5 + 165*x^6 + 572*x^7 + ...

Cf. A120010 (composition transpose), A000108 (Catalan), A000245.

Cf. A003517 (|a(n+1)|-|a(n)|). - Olivier Gérard, Oct 11 2012

[3*Catalan(n) - Catalan(n+1): n in [1..30]]; // Vincenzo Librandi, Jan 02 2025

f[n_]:=3*CatalanNumber[n] -CatalanNumber[n+1];Array[f,30,1] (* Vincenzo Librandi, Jan 02 2025 *)

a(n)=binomial(2*n,n)/(n+1)-4*binomial(2*n-1,n-2)/(n+2)

G.f.: A(x) = ((1-3*x)*sqrt(1-4*x) - (1-x)*(1-4*x))/(2*x^2) = x*C(x)^2 - x^2*C(x)^4 where C(x) is the Catalan function (A000108).
a(n) = C(2*n,n)/(n+1) - 4*C(2*n-1,n-2)/(n+2).
a(n) = 3*Catalan(n) - Catalan(n+1). - David Callan, Nov 21 2006
D-finite with recurrence: (n+2)*a(n) + (-7*n-2)*a(n-1) + 6*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Jan 20 2020, corrected Feb 16 2020
From Peter Bala, Feb 02 2024: (Start)
a(n) = 3*(-1)^n*Sum_{k = 0..n} (-4)^(n-k)*binomial(n,k)*(2*k + 2)!/((k + 3)!*k!).
G.f.: x/(1 - 4*x)*c(-x/(1 - 4*x))^3, where c(x) = (1 - sqrt(1 - 4*x))/(2*x) is the g.f. of the Catalan numbers A000108. (End)
E.g.f.: exp(2*x)*(2*BesselI(0, 2*x) - 3*BesselI(1, 2*x) + BesselI(2, 2*x)) - 2. - Stefano Spezia, Dec 31 2024

A001089 Number of permutations of [n] containing exactly 2 increasing subsequences of length 3.

Original entry on oeis.org

0, 0, 0, 0, 3, 24, 133, 635, 2807, 11864, 48756, 196707, 783750, 3095708, 12152855, 47500635, 185082495, 719559600, 2793121080, 10830450780, 41965864794, 162539516448, 629399492330, 2437072038302, 9437097796918

Offset: 0

John Thomas Noonan [ noonan(AT)euclid.math.temple.edu ]

nonn

			For n=4, there are 4! = 24 permutations of 1234. The identity permutation 1234 has four increasing subsequences of length 3 (123, 124, 134, and 234), and the permutation 2314 has only one increasing subsequence of length 3 (234). Only the permutations 1243, 1324, and 2134 have exactly two increasing subsequences of length 3, and since there are three of them, a(4) = 3. - _Michael B. Porter_, Sep 03 2016

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. Fulmek, Enumeration of permutations containing a prescribed number of occurrences of a pattern of length three, arXiv:math/0112092 [math.CO], 2001-2002. Also doi:10.1016/S0196-8858(02)00501-8, Adv. Appl. Math., 30, 2003, 607-632.
T. Mansour and A. Vainshtein, Counting occurrences of 123 in a permutation, arXiv:math/0105073 [math.CO], 2001.
Toufik Mansour, Sherry H. F. Yan and Laura L. M. Yang, Counting occurrences of 231 in an involution, Discrete Mathematics 306 (2006), pages 564-572.
J. Noonan and D. Zeilberger, The Enumeration of Permutations With a Prescribed Number of "Forbidden" Patterns, arXiv:math/9808080 [math.CO], 1998. Also doi:10.1006/aama.1996.0016, Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).

Cf. A003517, A084249, A138159.

Leading column of A229158.

Concatenation([0,0,0,0], List([4..30], n-> (100+117*n+59*n^2)* Binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)))); # G. C. Greubel, Sep 19 2019

[0,0,0,0] cat [(100+117*n+59*n^2)*Binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)): n in [4..30]]; // G. C. Greubel, Sep 19 2019

seq(`if`(n=0, 0, (100+117*n+59*n^2)*binomial(2*n, n-4)/(2*n*(2*n-1)*(n+5))), n = 0..30); # G. C. Greubel, Sep 19 2019

{0}~Join~CoefficientList[Series[((x^5-3x^4+5x^3-10x^2+6*x-1)(1-4x)^(1/2) - 5x^5+7x^4-17x^3+20x^2-8*x+1)/(2x^6), {x,0,23}], x] (* or *)
{0}~Join~CoefficientList[Series[x^5*((1-(1-4x)^(1/2))/(2x))^11 +3x^3*( (1-(1-4x)^(1/2))/(2x))^8, {x,0,23}], x] (* Michael De Vlieger, Sep 03 2016 *)

a(n) = (100+117*n+59*n^2)*binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)) \\ G. C. Greubel, Sep 19 2019

[0,0,0,0]+[(100+117*n+59*n^2)*binomial(2*n,n-4)/(2*n*(2*n-1)*(n+5)) for n in (4..30)] # G. C. Greubel, Sep 19 2019

Noonan and Zeilberger conjectured that a(n) = ((59*n^2+117*n+100) /(2*n*(2*n-1)*(n+5))) *binomial(2*n,n-4). This was proved by Fulmek.
G.f.: ((x^5 -3*x^4 +5*x^3 -10*x^2 +6*x -1)*(1-4*x)^(1/2) - 5*x^5 +7*x^4 -17*x^3 +20*x^2 -8*x +1)/(2*x^6). - Mark van Hoeij, Oct 25 2011
G.f.: x^5*C(x)^11 + 3*x^3*C(x)^8, where C(x) is g.f. for the Catalan numbers (A000108). - Michael D. Weiner, Sep 02 2016
D-finite with recurrence -(n+5)*(n-4)*(59*n^2-n+42)*a(n) +2*(n-1)*(2*n-3)*(59*n^2 +117*n+100)*a(n-1) = 0, equivalent to recurrence of [Noonan-Zeilberger] binomial. - R. J. Mathar, Jan 04 2017

Terms a(25) onward added by G. C. Greubel, Sep 19 2019

A145597 Generalized Narayana numbers, T(n, k) = 3/(n + 1)binomial(n + 1, k + 2)binomial(n + 1, k - 1), triangular array read by rows.

Original entry on oeis.org

1, 3, 3, 6, 15, 6, 10, 45, 45, 10, 15, 105, 189, 105, 15, 21, 210, 588, 588, 210, 21, 28, 378, 1512, 2352, 1512, 378, 28, 36, 630, 3402, 7560, 7560, 3402, 630, 36, 45, 990, 6930, 20790, 29700, 20790, 6930, 990, 45, 55, 1485, 13068, 50820, 98010, 98010

Offset: 2

Peter Bala, Oct 15 2008

T(n,k) is the number of walks of n unit steps, each step in the direction either up (U), down (D), right (R) or left (L), starting from (0,0) and finishing at lattice points on the horizontal line y = 2 and which remain in the upper half-plane y >= 0. An example is given in the Example section below.
The current array is the case r = 2 of the generalized Narayana numbers N_r(n,k) := (r + 1)/(n + 1)*binomial(n + 1,k + r)*binomial(n + 1,k - 1), which count walks of n steps from the origin to points on the horizontal line y = r that remain in the upper half-plane. Case r = 0 gives the table of Narayana numbers A001263 (but with an offset of 0 in the row numbering). For other cases see A145596 (r = 1), A145598 (r = 3) and A145599 (r = 4).
T(n,k) is the number of preimages of the permutation 3214567...(n+3) under West's stack-sorting map that have exactly k+1 descents. - Colin Defant, Sep 15 2018

			Triangle starts
n\k|..1.....2....3.....4.....5.....6
====================================
.2.|..1
.3.|..3.....3
.4.|..6....15....6
.5.|.10....45...45....10
.6.|.15...105..189...105....15
.7.|.21...210..588...588...210....21
...
Row 4: T(4,1) = 6: the 6 walks of length 4 from (0,0) to (-2,2) are LLUU, LULU, LUUL, ULLU, ULUL and UULL. Changing L to R in these walks gives the 6 walks from (0,0) to (2,2).
T(4,2) = 15: the 15 walks of length 4 from (0,0) to (0,2) are UUUD, UULR, UURL, UUDU,URUL, ULUR, URLU, ULRU, RUUL, LUUR, RLUU, LRUU, RULU, LURU and UDUU.
.
.
*......*......*......y......*......*......*
.
.
*......6......*.....15......*......6......*
.
.
*......*......*......*......*......*......*
.
.
*......*......*......o......*......*......* x axis
.

Harvey P. Dale, Table of n, a(n) for n = 2..1000
F. Cai, Q.-H. Hou, Y. Sun, and A. L. B. Yang, Combinatorial identities related to 2x2 submatrices of recursive matrices, arXiv:1808.05736 [math.CO], 2018; Table 2.1 for k=2.
Colin Defant, Preimages under the stack-sorting algorithm, arXiv:1511.05681 [math.CO], 2015-2018; Graphs Combin., 33 (2017), 103-122.
Colin Defant, Stack-sorting preimages of permutation classes, arXiv:1809.03123 [math.CO], 2018.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Cf. A003517 (row sums), A001263, A145596, A145598, A145599, A145601.

with(combinat):
T:= (n,k) -> 3/(n+1)*binomial(n+1,k+2)*binomial(n+1,k-1):
for n from 2 to 11 do
seq(T(n,k),k = 1..n-1);
end do;

Table[3/(n+1) Binomial[n+1,k+2]Binomial[n+1,k-1],{n,2,20},{k,n-1}]//Flatten (* Harvey P. Dale, Aug 12 2023 *)

T(n,k) = (3/(n+1))*binomial(n+1,k+2)*binomial(n+1,k-1) for n >=2 and 1 <= k <= n-1. In the notation of [Guy], T(n,k) equals w_n(x,y) at (x,y) = (2*k - n,2). Row sums A003517.
O.g.f. for column k+2: 3/(k + 1) * y^(k+3)/(1 - y)^(k+5) * Jacobi_P(k,3,1,(1 + y)/(1 - y)).
Identities for row polynomials R_n(x) := sum {k = 1..n-1} T(n,k)*x^k:
x^2*R_(n-1)(x) = 3*(n-1)*(n-2)/((n+1)*(n+2)*(n+3)) * Sum_{k = 0..n} binomial(n + 3,k) * binomial(2n - k,n) * (x - 1)^k;
Sum_{k = 1..n} (-1)^k*binomial(n,k)*R_k(x^2)*(1 + x)^(2*(n-k)) = R_n(1)*x^n = 6/(n+4)*binomial(2n+1,n-2)*x^n = A003517(n)*x^n.
Row generating polynomial R_(n+2)(x) = 3/(n+3)*x*(1-x)^n * Jacobi_P(n,3,3,(1+x)/(1-x)). [Peter Bala, Oct 31 2008]
G.f.: x*y*A001263(x,y)^3. - Vladimir Kruchinin, Nov 14 2020

A115145 Sixth convolution of A115140.

Original entry on oeis.org

1, -6, 9, -2, 0, 0, -1, -6, -27, -110, -429, -1638, -6188, -23256, -87210, -326876, -1225785, -4601610, -17298645, -65132550, -245642760, -927983760, -3511574910, -13309856820, -50528160150, -192113383644, -731508653106, -2789279908316, -10649977831752, -40715807302800

Offset: 0

Wolfdieter Lang, Jan 13 2006

Seiichi Manyama, Table of n, a(n) for n = 0..1669

Cf. A115139, A115140, A115141, A115142, A115143.

Cf. A115144, A115146, A115147, A115148, A115149.

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt(1-4*x))/2 )); // G. C. Greubel, Feb 12 2019

CoefficientList[Series[(1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*Sqrt[1-4*x])/2, {x, 0, 30}], x] (* G. C. Greubel, Feb 12 2019 *)

my(x='x+O('x^30)); Vec((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2) *sqrt(1-4*x))/2) \\ G. C. Greubel, Feb 12 2019

((1-6*x+9*x^2-2*x^3 +(1-4*x+3*x^2)*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 12 2019

O.g.f.: 1/c(x)^6 = P(7, x) - x*P(6, x)*c(x) with the o.g.f. c(x):=(1-sqrt(1-4*x))/(2*x) of A000108 (Catalan numbers) and the polynomials P(n, x) defined in A115139. Here P(7, x)=1-5*x+6*x^2-x^3 and P(6, x) = 1-4*x+3*x^2.
a(n) = -C6(n-6), n>=6, with C6(n) = A003517(n+2) (sixth convolution of Catalan numbers). a(0)=1, a(1)=-6, a(2)=9, a(3)=-2, a(4)=0=a(5). [1, -6, 9, -2] is row n=6 of signed A034807 (signed Lucas polynomials). See A115149 and A034807 for comments.
D-finite with recurrence +n*(n-6)*a(n) -2*(2*n-7)*(n-4)*a(n-1)=0. - R. J. Mathar, Sep 23 2021

A118919 Triangle read by rows: T(n,k) is the number of Grand Dyck paths of semilength n that cross downwards the x-axis k times. (A Grand Dyck path of semilength n is a path in the half-plane x>=0, starting at (0,0), ending at (2n,0) and consisting of steps u=(1,1) and d=(1,-1)).

Original entry on oeis.org

1, 2, 5, 1, 14, 6, 42, 27, 1, 132, 110, 10, 429, 429, 65, 1, 1430, 1638, 350, 14, 4862, 6188, 1700, 119, 1, 16796, 23256, 7752, 798, 18, 58786, 87210, 33915, 4655, 189, 1, 208012, 326876, 144210, 24794, 1518, 22, 742900, 1225785, 600875, 123970, 10350

Offset: 0

Emeric Deutsch, May 06 2006

Row n contains 1+floor(n/2) terms. Row sums are the central binomial coefficients (A000984). T(n,0)=A000108(n+1) (the Catalan numbers). T(n,1)=A003517(n). T(n,2)=A003519(n). Sum(k*T(n,k),k>=0)=A008549(n-1). For both downward and upward crossings, see A118920.
Eigenvector is defined by: A119243(n) = Sum_{k=0..[n\2]} T(n,k)*A119243(k). This triangle is closely related to triangle A119245. - Paul D. Hanna, May 10 2006
Column k is the sum of columns 2k and 2k+1 of A039599. - Philippe Deléham, Nov 11 2008

			T(3,1)=6 because we have ud\dudu,ud\dduu,udud\du,uudd\du,ud\duud and duud\du (the downward crossings of the x-axis are shown by a back-slash \).
Triangle starts:
  1;
  2;
  5,1;
  14,6;
  42,27,1;
  132,110,10;

Cf. A000984, A000108, A003517, A003519, A008549, A118920.

Cf. A119243 (eigenvector), A119245 (variant).

T:=(n,k)->(2*k+1)*binomial(2*n+2,n-2*k)/(n+1): for n from 0 to 13 do seq(T(n,k),k=0..floor(n/2)) od; # yields sequence in triangular form

T(n,k)=if(n<2*k || k<0,0,(2*k+1)*binomial(2*n+2,n-2*k)/(n+1)) \\ Paul D. Hanna, May 10 2006

T(n,k)=(2k+1)binomial(2n+2,n-2k)/(n+1). G.f.=G(t,z)=C^2/(1-tz^2*C^4), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.

T(n,k)=A039599(n,2k)+A039599(n,2k+1). - Philippe Deléham, Nov 11 2008

A090749 a(n) = 12 * C(2n+1,n-5) / (n+7).

Original entry on oeis.org

1, 12, 90, 544, 2907, 14364, 67298, 303600, 1332045, 5722860, 24192090, 100975680, 417225900, 1709984304, 6962078952, 28192122176, 113649492522, 456442180920, 1827459250276, 7297426411968, 29075683360185, 115631433392020, 459124809056550, 1820529677650320, 7210477496434485

Offset: 5

Philippe Deléham, Feb 15 2004

Also a diagonal of A059365 and A009766. See also A000108, A002057, A003517, A003518, A003519.

Number of standard tableaux of shape (n+6,n-5). - Emeric Deutsch, May 30 2004

G. C. Greubel, Table of n, a(n) for n = 5..1000
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, Bounce statistics for rational lattice paths, arXiv:1707.09918 [math.CO], 2017, p. 9.
Richard K. Guy, Letter to N. J. A. Sloane, May 1990.
Richard K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article 00.1.6.
V. E. Hoggatt, Jr., 7-page typed letter to N. J. A. Sloane with suggestions for new sequences, circa 1977.

Cf. A001622, A009766, A039598, A059365, A214292.

Cf. A000108, A002057, A003517, A003518, A003519.

Table[12*Binomial[2*n + 1, n - 5]/(n + 7), {n,5,50}] (* G. C. Greubel, Feb 07 2017 *)

for(n=5,50, print1(12*binomial(2*n+1,n-5)/(n+7), ", ")) \\ G. C. Greubel, Feb 07 2017

a(n) = A039598(n, 5) = A033184(n+7, 12).
G.f.: x^5*C(x)^12 with C(x) g.f. of A000108(Catalan).
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=11, a(n-6)=(-1)^(n-11)*coeff(charpoly(A,x),x^11). - Milan Janjic, Jul 08 2010
a(n) = A214292(2*n,n-6) for n > 5. - Reinhard Zumkeller, Jul 12 2012
From Karol A. Penson, Nov 21 2016: (Start)
O.g.f.: z^5 * 4^6/(1+sqrt(1-4*z))^12.
Recurrence: (-4*(n-5)^2-58*n+80)*a(n+1)-(-n^2-6*n+27)*a(n+2)=0, a(0),a(1),a(2),a(3),a(4)=0,a(5)=1,a(6)=12, n>=5.
Asymptotics: (-903+24*n)*4^n*sqrt(1/n)/(sqrt(Pi)*n^2).
Integral representation as n-th moment of a signed function W(x) on x=(0,4),in Maple notation: a(n+5)=int(x^n*W(x),x=0..4),n=0,1,..., where W(x)=(256/231)*sqrt(4-x)*JacobiP(5, 1/2, 1/2, (1/2)*x-1)*x^(11/2)/Pi and JacobiP are Jacobi polynomials. Note that W(0)=W(4)=0. (End).
From Ilya Gutkovskiy, Nov 21 2016: (Start)
E.g.f.: 6*exp(2*x)*BesselI(6,2*x)/x.
a(n) ~ 3*2^(2*n+3)/(sqrt(Pi)*n^(3/2)). (End)
From Amiram Eldar, Sep 26 2022: (Start)
Sum_{n>=5} 1/a(n) = 88699/15120 - 71*Pi/(27*sqrt(3)).
Sum_{n>=5} (-1)^(n+1)/a(n) = 102638*log(phi)/(75*sqrt(5)) - 22194839/75600, where phi is the golden ratio (A001622). (End)

Missing term 113649492522 inserted by Ilya Gutkovskiy, Dec 07 2016

A084249 Triangle T(n,k) read by rows: permutations on 123...n with one abc pattern and no aj pattern with j<=k, n>2, k

Original entry on oeis.org

1, 6, 2, 27, 12, 3, 110, 55, 19, 4, 429, 229, 91, 27, 5, 1638, 912, 393, 136, 36, 6, 6188, 3549, 1614, 612, 191, 46, 7, 23256, 13636, 6447, 2601, 897, 257, 57, 8, 87210, 52020, 25332, 10695, 3951, 1260, 335, 69, 9, 326876, 197676, 98532

Offset: 3

Ralf Stephan, May 21 2003

See A228708 for further information.

			Full triangle begins:
0
0,0
0,0,0
1,1,0,0
6,6,2,0,0
27,27,12,3,0,0
110,110,55,19,4,0,0
429,429,229,91,27,5,0,0
1638,1638,912,393,136,36,6,0,0
6188,6188,3549,1614,612,191,46,7,0,0
23256,23256,13636,6447,2601,897,257,57,8,0,0
...

J. Noonan and D. Zeilberger, [math/9808080] The Enumeration of Permutations With a Prescribed Number of ``Forbidden'' Patterns. Also Adv. in Appl. Math. 17 (1996), no. 4, 381--407. MR1422065 (97j:05003).

See A228708 for the full triangle.

T(n, 1) = A003517(n+1). Cf. A001089.

for(n=1,15, for(k=1,n-2,print1(binomial(2*n-k-1,n)-binomial(2*n-k-1,n+3)+binomial(2*n-2*k-2,n-k-4)-binomial(2*n-2*k-2,n-k-1)+binomial(2*n-2*k-3,n-k-4)-binomial(2*n-2*k-3,n-k-2)",")))

T(n, k) = C(2n-k-1, n) - C(2n-k-1, n+3) + C(2n-2k-2, n-k-4) - C(2n-2k-2, n-k-1) + C(2n-2k-3, n-k-4) - C(2n-2k-3, n-k-2).

T(n, n-2) = n-2, T(n, k) = T(n, k+1) + T(n-1, k-1) + T(n-k, 2).

cp's OEIS Frontend

A050144 T(n,k) = M(2n-1,n-1,k-1), 0 <= k <= n, n >= 0, where M(p,q,r) is the number of upright paths from (0,0) to (p,p-q) that meet the line y = x+r and do not rise above it.

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A138159 Triangle read by rows: T(n,k) is the number of permutations of [n] having k occurrences of the pattern 321 (n>=1, 0<=k<=n(n-1)(n-2)/6).

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A052720 Expansion of e.g.f.: (1 - 4*x + 3*x^2)*(1 - 2*x - sqrt(1-4*x))/2 - x^2 + 2*x^3.

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A120009 G.f.: A(x) = (x-x^2) o x/(1-x) o (1-sqrt(1-4*x))/2, a composition of functions involving the Catalan function and its inverse.

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A001089 Number of permutations of [n] containing exactly 2 increasing subsequences of length 3.

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A145597 Generalized Narayana numbers, T(n, k) = 3/(n + 1)*binomial(n + 1, k + 2)*binomial(n + 1, k - 1), triangular array read by rows.

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A115145 Sixth convolution of A115140.

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A052720 Expansion of e.g.f.: (1 - 4x + 3x^2)(1 - 2x - sqrt(1-4x))/2 - x^2 + 2x^3.

A145597 Generalized Narayana numbers, T(n, k) = 3/(n + 1)binomial(n + 1, k + 2)binomial(n + 1, k - 1), triangular array read by rows.