A018224 -id:A018224 - cp's OEIS frontend

A154413 a(n) = A006551(n) - A018224(n).

0, 0, 0, 2, 30, 202, 2016, 14394, 151290, 1294478, 15660744, 162298842, 2274318228, 27968231436, 447527038848, 6382757516250, 114890215021650, 1865385066804550, 37307710791708600, 679562209260462054

Offset: 0

Roger L. Bagula, Jan 09 2009

nonn

G. C. Greubel, Table of n, a(n) for n = 0..300

Cf. A006551, A018224.

t[n_, k_] = Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
a0 = Table[t[n, Floor[n/2]], {n, 1, 30}];
b0 = Table[Binomial[n, Floor[(n)/2]]^2, {n, 0, 29}];
a=a0-b0

a(n) = Eulerian[n,Floor[n/2]] - Binomial[n,Floor[n/2]]^2.

A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 3, 9, 9, 9, 3, 1, 1, 4, 12, 18, 18, 12, 4, 1, 1, 4, 16, 24, 36, 24, 16, 4, 1, 1, 5, 20, 40, 60, 60, 40, 20, 5, 1, 1, 5, 25, 50, 100, 100, 100, 50, 25, 5, 1, 1, 6, 30, 75, 150, 200, 200, 150, 75, 30, 6, 1, 1, 6, 36, 90, 225, 300, 400, 300, 225, 90, 36, 6, 1

Offset: 1

Emeric Deutsch, Nov 24 2003

Rows 2, 4, 6, ... give A088459.
Diagonal sums are in A088518(n-1). - Philippe Deléham, Jan 04 2009
Row sums are in A001405(n). - Philippe Deléham, Jan 04 2009
Subtriangle (1 <= k <= n) of triangle T(n,k), 0 <= k <= n, read by rows, given by A101455 DELTA A056594 := [0,1,0,-1,0,1,0,-1,0,1,0,-1,0,...] DELTA [1,0,-1,0,1,0,-1,0,1,0,-1,0,1,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
Also, number of symmetric noncrossing partitions of an n-set with k blocks. - Andrew Howroyd, Nov 15 2017
From Roger Ford, Oct 17 2018: (Start)
T(n,k) = t(n+2,d) where t(n,d) is the number of different semi-meander arch depth listings with n top arches and with d the depth of the deepest embedded arch.
Examples:    /\         semi-meander with 5 top arches
            //\\   /\   2 arches are at depth=0 (no covering arches)
           ///\\\ //\\  2 arches are at depth=1 (1 covering arch)
      (0)(1)(2)         1 arch is at depth=2 (2 covering arches)
       2, 2, 1 is the listing for this t(5,2)
             /\      semi-meander with 5 top arches
            /  \     (0)(1)
     /\ /\ //\/\\     3, 2  is the listing for this t(5,1)
a(6,5) = t(8,5)= 3 {2,1,1,1,2,1; 2,1,2,1,1,1; 3,1,1,1,1,1} (End)

			Triangle begins:
  1;
  1,  1;
  1,  1,  1;
  1,  2,  2,   1;
  1,  2,  4,   2,   1;
  1,  3,  6,   6,   3,    1;
  1,  3,  9,   9,   9,    3,    1;
  1,  4, 12,  18,  18,   12,    4,    1;
  1,  4, 16,  24,  36,   24,   16,    4,    1;
  1,  5, 20,  40,  60,   60,   40,   20,    5,    1;
  1,  5, 25,  50, 100,  100,  100,   50,   25,    5,    1;
  1,  6, 30,  75, 150,  200,  200,  150,   75,   30,    6,   1;
  1,  6, 36,  90, 225,  300,  400,  300,  225,   90,   36,   6,   1;
  1,  7, 42, 126, 315,  525,  700,  700,  525,  315,  126,  42,   7,  1;
  1,  7, 49, 147, 441,  735, 1225, 1225, 1225,  735,  441, 147,  49,  7, 1;
  1,  8, 56, 196, 588, 1176, 1960, 2450, 2450, 1960, 1176, 588, 196, 56, 8, 1;
  ...
a(6,2)=3 because we have UUUDDDUUUDDD, UUUUDDUUDDDD, UUUUUDUDDDDD, where
U=(1,1), D=(1,-1).

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Per Alexandersson, Svante Linusson, Samu Potka, and Joakim Uhlin, Refined Catalan and Narayana cyclic sieving, arXiv:2010.11157 [math.CO], 2020.
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Hyunsoo Cho, JiSun Huh, and Jaebum Sohn, The (s, s + d, ..., s + pd)-core partitions and the rational Motzkin paths, arXiv:2001.06651 [math.CO], 2020.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015-2016.
Johann Cigler, Pascal triangle, Hoggatt matrices, and analogous constructions, arXiv:2103.01652 [math.CO], 2021.
Nicolas Crampe, Julien Gaboriaud, and Luc Vinet, Racah algebras, the centralizer Z_n(sl_2) and its Hilbert-Poincaré series, arXiv:2105.01086 [math.RT], 2021.
L. Poulain d'Andecy, Centralisers and Hecke algebras in Representation Theory, with applications to Knots and Physics, arXiv:2304.00850 [math.RT], 2023. See p. 64.
Vladimir Shevelev, Several remarks on A088855, Seqfan thread, Nov 19 2017.

Cf. A005566, A018224, A056594, A084938, A088459, A101455, A209612, A247644.
Cf. A001405 (row sums), A088459, A088518 (diagonal sums).
Column 2 is A008619, column 3 is A002620, column 4 is A028724, column 5 is A028723, column 6 is A028725, column 7 is A331574.

[(&*[Binomial(Floor((n-j)/2), Floor((k-j)/2)): j in [0..1]]): k in [1..n], n in [1..15]]; // G. C. Greubel, Apr 08 2022

T[n_, k_] := Binomial[Quotient[n-1, 2], Quotient[k-1, 2]]*Binomial[ Quotient[n, 2], Quotient[k, 2]];
Table[T[n, k], {n,13}, {k,n}]//Flatten (* Jean-François Alcover, Jun 07 2018 *)

T(n,k) = binomial((n-1)\2, (k-1)\2)*binomial(n\2, k\2); \\ Andrew Howroyd, Nov 15 2017

def A088855(n,k): return product(binomial( (n-j)//2, (k-j)//2 ) for j in (0..1))
flatten([[A088855(n,k) for k in (1..n)] for n in (1..15)]) # G. C. Greubel, Apr 08 2022

T(n, k) = binomial(floor(n'), floor(k'))*binomial(ceiling(n'), ceiling(k')), where n' = (n-1)/2, k' = (k-1)/2.
G.f.: 2*u/(u*v + sqrt(x*y*u*v)) - 1, where x = 1+z+t*z, y = 1+z-t*z, u = 1-z+t*z, v = 1-z-t*z.
Triangle T(n,k), 0 <= k <= n, given by A101455 DELTA A056594 begins: 1; 0,1; 0,1,1; 0,1,1,1; 0,1,2,2,1; 0,1,2,4,2,1; 0,1,3,6,6,3,1; 0,1,3,9,9,9,3,1; ... - Philippe Deléham, Jan 03 2009
From G. C. Greubel, Apr 08 2022: (Start)
T(n, n-k+1) = T(n, k).
T(2*n-1, n) = A018224(n-1), n >= 1.
T(2*n, n) = A005566(n-1), n >= 1. (End)

Keyword:tabl added Philippe Deléham, Jan 25 2010

A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.

Original entry on oeis.org

1, 3, 2, 13, 16, 106, 166, 1073, 1934, 12142, 24076, 147090, 312906, 1865772, 4191822, 24463905, 57433950, 328887346, 800740450, 4508608610, 11319707546, 62781858592, 161841539812, 885513974674, 2335765140994, 12624162072740, 33979681977530, 181611275997040

Offset: 1

N. J. A. Sloane, Apr 10 2001

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Roland Bacher, Meander algebras

Meander sequences in Bacher's paper: A005315, A060066, A060089, A060111, A060148, A060149, A060174, A060198, A060206.

Cf. A018224.

seq(n) = Vec(1 - 1/sum(k=0, n, binomial(k, k\2)^2*x^k, O(x*x^n))) \\ Andrew Howroyd, Feb 07 2025

G.f.: 1 - 1/B(x) where B(x) is the g.f. of A018224. - Andrew Howroyd, Feb 07 2025

a(11) onwards from Andrew Howroyd, Feb 07 2025

A360859 Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).

Original entry on oeis.org

1, 1, 1, 1, 2, 4, 1, 3, 9, 9, 1, 4, 16, 24, 36, 1, 5, 25, 50, 100, 100, 1, 6, 36, 90, 225, 300, 400, 1, 7, 49, 147, 441, 735, 1225, 1225, 1, 8, 64, 224, 784, 1568, 3136, 3920, 4900, 1, 9, 81, 324, 1296, 3024, 7056, 10584, 15876, 15876, 1, 10, 100, 450, 2025, 5400, 14400, 25200, 44100, 52920, 63504

Offset: 0

Peter Luschny, Feb 28 2023

			Triangle T(n, k) starts:
[0] 1;
[1] 1, 1;
[2] 1, 2,  4;
[3] 1, 3,  9,   9;
[4] 1, 4, 16,  24,   36;
[5] 1, 5, 25,  50,  100,  100;
[6] 1, 6, 36,  90,  225,  300,  400;
[7] 1, 7, 49, 147,  441,  735, 1225,  1225;
[8] 1, 8, 64, 224,  784, 1568, 3136,  3920,  4900;
[9] 1, 9, 81, 324, 1296, 3024, 7056, 10584, 15876, 15876;

Cf. A018224 (main diagonal), A360861 (row sums).

Cf. A360857, A360858.

A360859 := (n, k) -> binomial(n, ceil(k/2)) * binomial(n, floor(k/2)):
seq(seq(A360859(n, k), k = 0..n), n = 0..10);

from math import comb
def A360859_T(n,k): return comb(n,m:=k>>1)**2*(n-m)//(m+1) if k&1 else comb(n,k>>1)**2 # Chai Wah Wu, Feb 28 2023

A113182 Number of unrooted two-vertex (or, dually, two-face) regular planar maps of valency n considered up to orientation-preserving homeomorphism.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 39, 95, 308, 859, 3013, 9130, 33300, 106039, 394340, 1297295, 4878109, 16428300, 62232321, 213388961, 812825244, 2827645453, 10818489817, 38086408002, 146250545528, 520062618300, 2003199281223, 7184570776213

Offset: 1

Valery A. Liskovets, Oct 19 2005

nonn

Bisections are A112944 and A113181.

M. Bousquet, G. Labelle and P. Leroux, Enumeration of planar two-face maps, Discrete Math., vol. 222 (2000), 1-25.

Cf. A018224, A112944, A113181.

a[n_] := If[OddQ[n], (1/(2n))(Sum[EulerPhi[d] Binomial[2 Floor[(n-1)/(2d)], Floor[(n-1)/(2d)]]^2, {d, Divisors[n]}] + n Binomial[n-1, (n-1)/2]), (1/4)((2 Sum[EulerPhi[d] Binomial[n/d-1, Floor[n/(2d)]]^2, {d, Divisors[ n]}])/n + Binomial[n, n/2])];
Array[a, 28] (* Jean-François Alcover, Aug 30 2019 *)

A378061 Triangle read by rows: T(n, k) = binomial(n + 1, (n - k)/2)^2*(k + 1)/(n + 1) if n - k is even, otherwise 0.

Original entry on oeis.org

1, 0, 1, 3, 0, 1, 0, 8, 0, 1, 20, 0, 15, 0, 1, 0, 75, 0, 24, 0, 1, 175, 0, 189, 0, 35, 0, 1, 0, 784, 0, 392, 0, 48, 0, 1, 1764, 0, 2352, 0, 720, 0, 63, 0, 1, 0, 8820, 0, 5760, 0, 1215, 0, 80, 0, 1, 19404, 0, 29700, 0, 12375, 0, 1925, 0, 99, 0, 1

Offset: 0

Peter Luschny, Dec 07 2024

Consider square lattice walks with unit steps in all four directions (NSWE), starting at the origin, ending on the y-axis, and never going below the x-axis. T(n, k) is the number of walks with length n and height k. The number of walks with positive height is A378060, and with nonnegative height is A018224. Walks of odd length can never have an even height, and walks of even length cannot have an odd height. The Python program below generates the walks.

			Triangle starts:
  0  [   1]
  1  [   0,    1]
  2  [   3,    0,    1]
  3  [   0,    8,    0,    1]
  4  [  20,    0,   15,    0,   1]
  5  [   0,   75,    0,   24,   0,    1]
  6  [ 175,    0,  189,    0,  35,    0,  1]
  7  [   0,  784,    0,  392,   0,   48,  0,  1]
  8  [1764,    0, 2352,    0, 720,    0, 63,  0, 1]
  9  [   0, 8820,    0, 5760,   0, 1215,  0, 80, 0, 1]
.
The 15 walks with length 4 and height 2 are: 'NNNS', 'NNSN', 'NNWE', 'NNEW', 'NSNN', 'NWNE', 'NWEN', 'NENW', 'NEWN', 'WNNE', 'WNEN', 'WENN', 'ENNW', 'ENWN', 'EWNN'.

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

The columns are aerated rows of A378062. See also: A000891, A145600, A145601, A145602, A145603.

Cf. A018224 (row sums), A378060.

T := (n, k) -> ifelse((n - k)::odd, 0, binomial(n+1, (n-k)/2)^2*(k+1)/(n+1)):
for n from 0 to 9 do seq(T(n, k), k = 0..n) od;

T[n_, k_] := If[EvenQ[n-k],Binomial[n + 1, (n - k)/2]^2*(k + 1)/(n + 1), 0]; Table[T[n,k],{n,0,10},{k,0,n}]//Flatten (* Stefano Spezia, Dec 08 2024 *)

# Creates the table by counting the heights of square lattice walks. For illustration only.
from dataclasses import dataclass
@dataclass
class Z: w: str = ""; r: int = 0; i: int = 0
def Trow(n: int) -> list[int]:
    W = [Z()]
    row = [0] * (n + 1)
    for x in W:
        if len(x.w) == n:
            if x.r == 0: row[x.i] += 1
        else:
            for s in "NSWE":
                r = i = 0
                match s:
                    case "W": r = 1
                    case "E": r = -1
                    case "N": i = 1
                    case "S": i = -1
                if x.i + i >= 0:
                    W.append(Z(x.w + s, x.r + r, x.i + i))
    return row
for n in range(10): print(f"[{n}] {Trow(n)}")

A378067 Triangle read by rows: T(n, k) is the number of walks of length n with unit steps in all four directions (NSWE) starting at (0, 0), staying in the upper plane (y >= 0), and ending on the vertical line x = 0 if k = 0, or on the line x = k or x = -(n + 1 - k) if k > 0.

Original entry on oeis.org

1, 1, 2, 4, 3, 3, 9, 10, 6, 10, 36, 25, 20, 20, 25, 100, 101, 55, 50, 55, 101, 400, 301, 231, 126, 126, 231, 301, 1225, 1226, 742, 490, 294, 490, 742, 1226, 4900, 3921, 3144, 1632, 1008, 1008, 1632, 3144, 3921, 15876, 15877, 10593, 7137, 3348, 2592, 3348, 7137, 10593, 15877

Offset: 0

Peter Luschny, Dec 08 2024

			Triangle starts:
  [0] [    1]
  [1] [    1,     2]
  [2] [    4,     3,     3]
  [3] [    9,    10,     6,   10]
  [4] [   36,    25,    20,   20,   25]
  [5] [  100,   101,    55,   50,   55,  101]
  [6] [  400,   301,   231,  126,  126,  231,  301]
  [7] [ 1225,  1226,   742,  490,  294,  490,  742, 1226]
  [8] [ 4900,  3921,  3144, 1632, 1008, 1008, 1632, 3144,  3921]
  [9] [15876, 15877, 10593, 7137, 3348, 2592, 3348, 7137, 10593, 15877]
.
For n = 3 we get the walks depending on the x-coordinate of the endpoint:
W(x= 3) = {WWW},
W(x= 2) = {NWW,WNW,WWN},
W(x= 1) = {NNW,NSW,NWN,NWS,WWE,WEW,EWW,WNN,WNS},
W(x= 0) = {NNN,NNS,NSN,NWE,NEW,WNE,WEN,ENW,EWN},
W(x=-1) = {NNE,NEN,ENN,NSE,NES,WEE,ENS,EWE,EEW},
W(x=-2) = {NEE,ENE,EEN},
W(x=-3) = {EEE}.
T(3, 0) = card(W(x=0)) = 9, T(3, 1) = card(W(x=1)) + card(W(x=-3)) = 10,
T(3, 2) = card(W(x=2)) + card(W(x=-2)) = 6, T(3, 3) = card(W(x=3)) + card(W(x=-1)) = 10.

Alin Bostan, Computer Algebra for Lattice Path Combinatorics, Séminaire de Combinatoire Philippe Flajolet, Institut Henri Poincaré, March 28, 2013.
Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008-2009; Discrete Mathematics & Theoretical Computer Science, DMTCS Proceedings vol. AK, (FPSAC 2009).
R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6.

Related triangles: A052174 (first quadrant), this triangle (upper plane), A379822 (whole plane).

Cf. A018224 (column 0), A001700 (row sums), A378069 (row sum without column 0), A380121 (row minimum).

from dataclasses import dataclass
@dataclass
class Walk:
    s: str = ""
    x: int = 0
    y: int = 0
def Trow(n: int) -> list[int]:
    W = [Walk()]
    row = [0] * (n + 1)
    for w in W:
        if len(w.s) == n:
            row[w.x] += 1
        else:
            for s in "NSWE":
                x = y = 0
                match s:
                    case "W": x =  1
                    case "E": x = -1
                    case "N": y =  1
                    case "S": y = -1
                    case _  : pass
                if w.y + y >= 0:
                    W.append(Walk(w.s + s, w.x + x, w.y + y))
    return row
for n in range(10): print(Trow(n))

Sum_{k=1..n} T(n, k) = 2 * A378069(n).

A360861 a(n) = Sum_{k=0..n} binomial(n, ceiling(k/2)) * binomial(n, floor(k/2)).

Original entry on oeis.org

1, 2, 7, 22, 81, 281, 1058, 3830, 14605, 54127, 208110, 782761, 3027038, 11501478, 44668692, 170974710, 666220005, 2564271875, 10018268150, 38728479647, 151631858378, 588229029258, 2307174835212, 8975958379817, 35258881445606, 137501193282026, 540821096592028

Offset: 0

Peter Luschny, Feb 28 2023

nonn

Paolo Xausa, Table of n, a(n) for n = 0..1000

Row sums of A360859.

Cf. A001405, A018224.

A360861[n_]:=(Binomial[2n+1,n]+Binomial[n,Floor[n/2]]^2)/2;
Array[A360861,30,0] (* Paolo Xausa, Dec 11 2023 *)

a(n):=(1/2)*(binomial(2*n+1,n)+(binomial(n,floor(n/2)))^2); /* Tani Akinari, Jul 12 2023 */

from math import comb
def A360861(n): return sum(comb(n,m:=k>>1)**2*(n-m)//(m+1) for k in range(1,n+1,2)) + sum(comb(n,k>>1)**2 for k in range(0,n+1,2)) # Chai Wah Wu, Feb 28 2023

a(n) = (1/2)*(binomial(2*n+1,n)+binomial(n,floor(n/2))^2). - Tani Akinari, Jul 12 2023

A302182 Number of 3D walks of type abc.

Original entry on oeis.org

1, 1, 5, 12, 62, 200, 1065, 3990, 21714, 89082, 492366, 2147376, 12004740, 54718092, 308559537, 1454116950, 8255788970, 39935276810, 227976044010, 1126178350440, 6457854821340, 32456552441040, 186814834574550, 952569927106980, 5500292590186380, 28391993275117500

Offset: 0

N. J. A. Sloane, Apr 09 2018

See Dershowitz (2017) for precise definition.

Nachum Dershowitz, Touchard's Drunkard, Journal of Integer Sequences, Vol. 20 (2017), #17.1.5.

Cf. A000108, A000984, A002212, A002896, A005572, A026375, A064037, A081671, A138547, A145847, A145867, A150500, A202814.

Cf. A018224, A126120, A135394, A302181.

from math import comb as binomial
def row(n: int) -> list[int]:
    return sum(binomial(n, k)*binomial(k, k//2)//(k//2+1)*((k+1) %2)*binomial(n-k, (n-k)//2)**2 for k in range(n+1))
for n in range(26): print(row(n)) # Mélika Tebni, Nov 27 2024

From Mélika Tebni, Nov 27 2024: (Start)
a(n) = Sum_{k=0..n} binomial(n, k)*A126120(k)*A018224(n-k).
a(2*n+1) = A135394(n) / (2*n+2).
a(2*n) = A302181(n). (End)

a(13)-a(25) from Mélika Tebni, Nov 27 2024

A138354 Central moment sequence of tr(A^4) in USp(4).

Original entry on oeis.org

1, 0, 3, 1, 21, 26, 215, 498, 2821, 9040, 43695, 165375, 752785, 3101970, 13881803, 59837183, 267860685, 1184749704, 5337504263, 23996776941, 108964583121, 495544446410, 2267450194443, 10402298479276, 47926692348121

Offset: 0

Andrew V. Sutherland, Mar 16 2008, Mar 31 2008

nonn

Binomial transform of A018224.

If A is a random matrix in the compact group USp(4) (4 X 4 complex are unitary and symplectic), then a(n)=E[(tr(A^4)+1)^n] is the n-th central moment of the trace of A^4, since E[tr(A^4)] = -1 (see A018224).

			a(3) = 1 because E[(tr(A^4)+1)^3] = 1.
a(3) = 1*A018224(0) + 3*A018224(1) + 3*A018224(2) + 1*A018224(1) = 1*1 + 3*(-1) + 3*4 + 1*(-9) = 1.

Kiran S. Kedlaya and Andrew V. Sutherland, Hyperelliptic curves, L-polynomials and random matrices, arXiv:0803.4462 [math.NT], 2008-2010.

Cf. A018224.

a18224[n_] := Binomial[n, Floor[n/2]]^2;
a[n_] := Sum[(-1)^i Binomial[n, i] a18224[i], {i, 0, n}];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 13 2018 *)

a(n) = (1/2)*Integral_{x=0..Pi,y=0..Pi}(2cos(4x)+2cos(4y)+1)^n*(2cos(x)-2cos(y))^2*(2/Pi*sin^2(x))*(2/Pi*sin^2(y))dxdy.
a(n) = Sum_{i=0..n} (-1)^i binomial(n,i)*A018224(i). [corrected by Jean-François Alcover, Aug 13 2018]
a(n) = (5*A201805(n) - A201805(n+1))/4. - Mark van Hoeij, Nov 29 2024

cp's OEIS Frontend

A154413 a(n) = A006551(n) - A018224(n).

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A088855 Triangle read by rows: number of symmetric Dyck paths of semilength n with k peaks.

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Magma

Mathematica

PARI

Sage

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Extensions

A060149 Number of homogeneous generators of degree n for graded algebra associated with meanders.

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PARI

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A360859 Triangle read by rows. T(n, k) = binomial(n, ceil(k/2)) * binomial(n, floor(k/2)).

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Maple

Python

A113182 Number of unrooted two-vertex (or, dually, two-face) regular planar maps of valency n considered up to orientation-preserving homeomorphism.

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Mathematica

A378061 Triangle read by rows: T(n, k) = binomial(n + 1, (n - k)/2)^2*(k + 1)/(n + 1) if n - k is even, otherwise 0.

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Maple

Mathematica

Python

A378067 Triangle read by rows: T(n, k) is the number of walks of length n with unit steps in all four directions (NSWE) starting at (0, 0), staying in the upper plane (y >= 0), and ending on the vertical line x = 0 if k = 0, or on the line x = k or x = -(n + 1 - k) if k > 0.

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Python

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A360861 a(n) = Sum_{k=0..n} binomial(n, ceiling(k/2)) * binomial(n, floor(k/2)).

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