A000891 -id:A000891 - cp's OEIS frontend

A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

1, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 20, 20, 4, 1, 1, 70, 175, 50, 5, 1, 1, 252, 1764, 980, 105, 6, 1, 1, 924, 19404, 24696, 4116, 196, 7, 1, 1, 3432, 226512, 731808, 232848, 14112, 336, 8, 1, 1, 12870, 2760615, 24293412, 16818516, 1646568, 41580, 540, 9, 1

Offset: 0

Paul Barry, Jun 13 2006

Columns are the central coefficients of the triangles T(n, k;r) with T(n, k;r)=Product{j=0..r, C(n+j, k+j)/C(n-k+j, j)}*[k<=n]; (r=0,A007318), (r=1;A001263),(r=2,A056939),(r=3,A056940),(r=4,A056941). Essentially A103905 as a number triangle with an extra diagonal of 1's. Central coefficients T(2n, n) are A008793. Row sums are A120259. Diagonal sums are A120260.

			Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   6,     3,     1;
  1,  20,    20,     4,    1;
  1,  70,   175,    50,    5,   1;
  1, 252,  1764,   980,  105,   6, 1;
  1, 924, 19404, 24696, 4116, 196, 7, 1;
  ...

Seiichi Manyama, Rows n = 0..100, flattened

Row sums give A120259.

Cf. A000891, A000984, A103905, A120257.

T(n, k) = prod(j=0, k-1, binomial(2*n-2*k+j, n-k)/binomial(n-k+j, j)); \\ Seiichi Manyama, Apr 02 2021

Number triangle T(n, k)=[k<=n]*Product{j=0..k-1, C(2n-2k+j, n-k)/C(n-k+j, j)}

As a square array, this is T(n,m)=product{k=1..m, product{j=1..n, product{i=1..n, (i+j+k-1)/(i+j+k-2)}}}; - Paul Barry, May 13 2008

A038535 Numerators of coefficients of EllipticE/Pi.

Original entry on oeis.org

1, -1, -3, -5, -175, -441, -4851, -14157, -2760615, -8690825, -112285459, -370263621, -19870814327, -67607800225, -931331941875, -3241035157725, -2913690606794775, -10313859829588425, -147068001273760875, -527570807893408125, -30451387031607516975

Offset: 0

Wouter Meeussen, revised Jan 03 2001

Contribution from Wolfdieter Lang, Nov 08 2010: (Start)
a(n)/A056982(n) = -(binomial(2*n,n)^2)/((2*n-1)*2^(4*n)), n>=0, are the coefficients of x^n of hypergeometric([1/2,-1/2],[1],x).
The series hypergeometric([1/2,-1/2],[1],e^2)=L/(2*Pi*a) with L the perimeter of an ellipse with major axis a and numerical eccentricity e. (End)

Cf. A038533, A038534.

a(n) divides A000891(n+1).

Numerator[CoefficientList[Series[EllipticE[m]/Pi,{m,0,25}],m]] (* Harvey P. Dale, Dec 16 2011 *)

a(n) = 2^(-2 w[n])binomial[2n, n]^2 (-1)^(2n)/(1-2n) with w[n]=A000120 = number of 1's in binary expansion of n

A123619 a(n) = A123610(2*n+2,n)/(n+1) = A123618(n)/(n+1).

Original entry on oeis.org

1, 2, 13, 98, 884, 8712, 92033, 1022450, 11819620, 141052808, 1727897780, 21634496072, 275950213712, 3576314656800, 46995009879033, 625082413914450, 8403885788094500, 114069363868845000, 1561609591376307572

Offset: 0

Paul D. Hanna, Oct 03 2006

nonn

Related sequences: A123610(2n,n) = A123617(n); A123610(2n+1,n) = A000891(n); A123610(2n+2,n) = A123618(n).

G. C. Greubel, Table of n, a(n) for n = 0..830

Cf. A123610 (triangle); A123617, A000891, A123618.

T[, 0] = 1; T[n, k_] := 1/n DivisorSum[n, If[GCD[k, #] == #, EulerPhi[#]*Binomial[n/#, k/#]^2, 0] &];
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* A123610 *)
Table[T[2*n, n], {n, 0, 50}] (* A123617 *)
Table[T[2*n + 2, n], {n, 0, 50}] (* A123618 *)
Table[T[2*n + 2,n]/(n+1), {n, 0, 50}] (* A123619 *)
(* G. C. Greubel, Oct 26 2017 *)

{a(n)=if(n==0,1,(1/(2*(n+1)^2))*sumdiv(2*n+2,d,if(gcd(n,d)==d, eulerphi(d)*binomial((2*n+2)/d,n/d)^2,0)))}

A333667 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2n+2, read by rows, where T(n,k) is the number of 2(k+2*n-2)-cycles in the n X n grid graph which pass through NW and SE corners ((0,0),(n-1,n-1)).

Original entry on oeis.org

1, 3, 20, 16, 6, 175, 420, 562, 456, 186, 1764, 8064, 21224, 39500, 55376, 57248, 37586, 10260, 1072, 19404, 138600, 569768, 1717152, 4151965, 8371428, 14126846, 19364732, 20241450, 14759356, 6998166, 1927724, 230440

Offset: 2

Seiichi Manyama, Apr 01 2020

			T(3,0) = 3;
   +--*--*   +--*--*   +--*
   |     |   |     |   |  |
   *--*  *   *     *   *  *--*
      |  |   |     |   |     |
      *--+   *--*--+   *--*--+
Triangle starts:
=======================================================================
n\k|      0        1         2 ...      4 ...   8 ...    12 ...     18
---|-------------------------------------------------------------------
2  |      1;
3  |      3;
4  |     20,      16,        6;
5  |    175,     420,      562, ... , 186;
6  |   1764,    8064,    21224, .......... , 1072;
7  |  19404,  138600,   569768, .................. , 230440;
8  | 226512, 2265120, 12922446, ............................ , 4638576;

Seiichi Manyama, Rows n = 2..9, flattened

Row sums give A333323.

Cf. A003763, A302337, A333651, A333652, A333668.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A333667(n):
    universe = tl.grid(n - 1, n - 1)
    GraphSet.set_universe(universe)
    cycles = GraphSet.cycles().including(1).including(n * n)
    return [cycles.len(2 * k).len() for k in range(2 * n - 2, n * n // 2 + 1)]
print([i for n in range(2, 8) for i in A333667(n)])

T(n,0) = A000891(n-2).

A358553 Number of internal (non-leaf) nodes in the n-th standard ordered rooted tree.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 2, 3, 2, 2, 1, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 4, 3, 2, 5, 4, 4, 3, 3, 4, 4, 3, 4, 3, 3, 2, 4, 2, 4, 3, 4, 3, 3, 2, 2, 3, 3, 2, 3, 2, 2, 1, 3, 3, 5, 4, 4, 3, 3, 2, 4, 5, 5, 4, 5, 4, 4, 3, 5, 3, 5, 4, 5, 4, 4

Offset: 1

Gus Wiseman, Nov 26 2022

nonn

We define the n-th standard ordered rooted tree to be obtained by taking the (n-1)-th composition in standard order (graded reverse-lexicographic, A066099) as root and replacing each part with its own standard ordered rooted tree. This ranking is an ordered variation of Matula-Goebel numbers, giving a bijective correspondence between positive integers and unlabeled ordered rooted trees.

			The 89-th standard rooted tree is ((o)o(oo)), and it has 3 internal nodes, so a(89) = 3.

This statistic is counted by A001263, unordered A358575 (reverse A055277).
The unordered version is A342507, firsts A358554.
Other statistics: A358371 (leaves), A358372 (nodes), A358379 (edge-height).
A000081 counts rooted trees, ordered A000108.
Cf. A000891, A014221, A358373, A358585, A358586, A358588.

stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]];
srt[n_]:=If[n==1,{},srt/@stc[n-1]];
Table[Count[srt[n],[_],{0,Infinity}],{n,100}]

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)}")

A382136 Number of triples of non-crossing lattice paths from (0,0) to (n,n) using (1,0) and (0,1) as steps.

Original entry on oeis.org

1, 4, 50, 980, 24696, 731808, 24293412, 877262100, 33803832920, 1371597504992, 58043512597616, 2543610972177184, 114801908084920000, 5313688317073440000, 251370667949555421000, 12120154230252872020500, 594283640753967620247000, 29576997448419995135100000

Offset: 0

Yifan Xie, Mar 27 2025

a(n) is the number of triples (A, B, C) of paths having no common vertices and using (1,0) and (0,1) as steps, where A is from (0,0) to (n,n), B is from (1,-1) to (n+1,n-1), and C is from (2,-2) to (n+2,n-2).
a(n) is the number of ways to fill a n X n grid with numbers 1, 2, 4, 8 such that each number divides the number to the right and to the top.
a(n) is the number of secondary GL(3) invariants contructed from n+2 distinct three component vectors. This number was evaluated by using the Molien-Weyl formula to compute the Hilbert series of the ring of invariants. - Jaco van Zyl, Jun 30 2025

			For n = 2, the triple {NNEE, NENE, ENEN} is valid, while {ENNE, NNEE, NEEN} is invalid.

Paolo Xausa, Table of n, a(n) for n = 0..550
Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 9; Journal of High Energy Physics 06 (2025) 246.
Wikipedia, Lindstrom-Gessel-Viennot lemma

Cf. A000108, A000891.

A382136[n_] := If[n == 0, 1, 4*Binomial[2*n+1, n-1]^2*Binomial[2*n, n-1]/n^3];
Array[A382136, 20, 0] (* Paolo Xausa, Jul 03 2025 *)

a(n) = if(n==0, 1, 4*binomial(2*n+1, n-1)^2*binomial(2*n, n-1)/n^3)

From the Lindstrom-Gessel-Viennot lemma and using the definition from the first comment, a(n) is the determinant of the matrix:
  C(2*n, n)    C(2*n, n-1)  C(2*n, n-2)
  C(2*n, n+1)  C(2*n, n)    C(2*n, n-1)
  C(2*n, n+2)  C(2*n, n+1)  C(2*n, n)
a(n) = 4*C(2*n+1,n-1)*C(2*n+1,n+2)*C(2*n,n+1)/n^3 for n >= 1.

A120307 Inverse determinant of n X n matrix M[i,j] = i*j/(i+j-1).

Original entry on oeis.org

1, 3, 60, 10500, 18522000, 359400888000, 81408613942656000, 224737840779305293440000, 7812628980363223707442752000000, 3508978524227146242839564498172672000000

Offset: 1

Alexander Adamchuk, Jul 15 2006

nonn

Cf. A000108, A000891, A001700.

Table[ 1/Det[ Table[ i*j/(i+j-1), {i, n}, {j, n}]], {n,1,12}]

def A120307(n): return A163085(2*n)/factorial(2*n)
[A120307(n) for n in (1..10)] # Peter Luschny, Sep 18 2012

a(n) = 1/Det[ Table[ i*j/(i+j-1), {i, n}, {j, n}]]. a(n+1)/a(n) = A000891[n] = (2n)!(2n+1)! / (n! (n+1)!)^2 = (2n+1)*CatalanNumber[n]^2 = (2n+1)*A000108[n]^2 = C(2n+1,n+1)*CatalanNumber[n] = A001700[n]*A000108[n].

a(n) = A163085(2*n)/(2*n)!. - Peter Luschny, Sep 18 2012

A267981 a(n) = Catalan(n)^2*(4n + 2).

Original entry on oeis.org

2, 6, 40, 350, 3528, 38808, 453024, 5521230, 69526600, 898283672, 11848435872, 158966514616, 2163449607200, 29802622140000, 414852500188800, 5827381213589550, 82510878636707400, 1176544010190087000, 16882265852589060000, 243611096252860135800

Offset: 0

Ralf Steiner, Jan 23 2016

Numerator of (4n+2)*(Wallis-Lambert-series-1)(n) with denominator A013709(n) convergent to 2*(1-2/Pi). Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 2*(1-2/Pi). Q.E.D.

			For n=3 the a(3)=350.

Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).

Cf. A013709 (denominator). Equals twice A000891.

[Catalan(n)^2*(4*n+2):n in [0..20]]; // Vincenzo Librandi, Jan 25 2016

Table[CatalanNumber[n]^2 (4 n + 2), {n, 0, 20}] (* Vincenzo Librandi, Jan 25 2016 *)

a000108(n) = binomial(2*n, n)/(n+1)
a(n) = a000108(n)^2 * (4*n+2) \\ Felix Fröhlich, Jul 14 2016

G.f.: (Pi-2*EllipticE(16*x))/(2*Pi*x). - Benedict W. J. Irwin, Jul 14 2016
a(n) ~ 4^(2*n+1)/(Pi*n^2). - Ilya Gutkovskiy, Jul 14 2016
Recurrence: (n+1)^2*a(n) = 4*(2*n - 1)*(2*n + 1)*a(n-1). - Vaclav Kotesovec, Jul 16 2016
Sum_{n>=0} a(n)/2^(4*n+2) = 2 - 4/Pi. - Vaclav Kotesovec, Jul 16 2016

More terms from Vincenzo Librandi, Jan 25 2016

A362207 a(n) is the number of unordered triples of shortest nonintersecting grid paths joining two opposite corners of an n X n X n grid.

Original entry on oeis.org

2, 1440, 5039744, 30456915312, 244247250106272, 2330237215901633376, 25005390829898900970720, 292102859220245236374450192, 3638369778575244135648725730848, 47651985114895805442163075548018912, 649794504408024777960179124905242154688

Offset: 1

Janaka Rodrigo, Apr 11 2023

nonn

An n X n X n grid is a cube with (n+1)^3 grid points. Each of the three paths has 3*n steps and they intersect only at the start and at the end. Initially, each path will start by moving along distinct coordinate axes.

The two-dimensional version of this sequence is A000891(n-1).

a(4)-a(11) from Andrew Howroyd, Apr 11 2023

cp's OEIS Frontend

A120258 Triangle of central coefficients of generalized Pascal-Narayana triangles.

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A038535 Numerators of coefficients of EllipticE/Pi.

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A123619 a(n) = A123610(2*n+2,n)/(n+1) = A123618(n)/(n+1).

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A333667 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2*n+2, read by rows, where T(n,k) is the number of 2*(k+2*n-2)-cycles in the n X n grid graph which pass through NW and SE corners ((0,0),(n-1,n-1)).

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A358553 Number of internal (non-leaf) nodes in the n-th standard ordered rooted tree.

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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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A382136 Number of triples of non-crossing lattice paths from (0,0) to (n,n) using (1,0) and (0,1) as steps.

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A120307 Inverse determinant of n X n matrix M[i,j] = i*j/(i+j-1).

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A333667 Triangle T(n,k), n >= 2, 0 <= k <= floor(n^2/2)-2n+2, read by rows, where T(n,k) is the number of 2(k+2*n-2)-cycles in the n X n grid graph which pass through NW and SE corners ((0,0),(n-1,n-1)).