F. Chapoton - cp's OEIS frontend

A383776 a(n) = (11n + 3 + 6/(n+2)) Catalan(n).

6, 16, 53, 186, 672, 2472, 9207, 34606, 130988, 498576, 1906346, 7316596, 28170768, 108760560, 420889995, 1632155670, 6340808820, 24673450560, 96148670310, 375164728620, 1465589068320, 5731488987120, 22436098732710, 87905595401676, 344702077523352, 1352701532137312, 5312100899224532, 20874451526714856

Offset: 0

F. Chapoton, May 09 2025

It appears that for n >= 2 a(n-2) is the number of lattice points in the n-dimensional lattice polytope defined, in the space with coordinates (x_1,x_2,...,x_n), by the equations x_i >= 0 for every i, sum_i x_i <= n and x_1 + x_2 <= 2. For n=2, this is a triangle with 6 lattice points.

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

Cf. A000108, A000984, A007054, A051960.

A383776[n_] := (11*n + 3 + 6/(n + 2))*CatalanNumber[n];
Array[A383776, 30, 0] (* Paolo Xausa, May 15 2025 *)

[(11*n+3+6/(n+2))*catalan_number(n) for n in range(12)]

a(n) = (11*n + 3 + 6/(n + 2))*Catalan(n).

G.f.: 2*(7 + 5*sqrt(1 - 4*x) - 6*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)). - Stefano Spezia, May 15 2025

A382880 Symmetric triangle read by rows refining A109113.

Original entry on oeis.org

1, 1, 1, 6, 6, 1, 1, 11, 33, 33, 11, 1, 1, 16, 85, 189, 189, 85, 16, 1, 1, 21, 162, 590, 1107, 1107, 590, 162, 21, 1, 1, 26, 264, 1361, 3919, 6588, 6588, 3919, 1361, 264, 26, 1, 1, 31, 391, 2627, 10400, 25484, 39663, 39663, 25484, 10400, 2627, 391, 31, 1

Offset: 0

F. Chapoton, Apr 07 2025

Every row is symmetric. Alternating row sums vanish.

			Triangle begins:
  1,  1;
  1,  6,   6,   1;
  1, 11,  33,  33,   11,    1;
  1, 16,  85, 189,  189,   85,  16,   1;
  1, 21, 162, 590, 1107, 1107, 590, 162, 21, 1;
  ...

Cf. A109113 (row sums).

y = polygen(QQ,'y')
t = LazyPowerSeriesRing(y.parent(),'t').gen()
v = (1+y)*(1+y*t)/(1-(1+4*y+y**2)*t-(y+y**2+y**3)*t**2)
[list(cf) for cf in v[:8]]

G.f.: (1+y)*(1+y*t)/(1-(1+4*y+y^2)*t-(y+y^2+y^3)*t^2).

A382668 a(n) = C(n+1) - C(n-1) - 2*C(n-2) where C(n) = A000108(n) are the Catalan numbers.

Original entry on oeis.org

2, 10, 33, 108, 359, 1214, 4169, 14508, 51064, 181492, 650522, 2348856, 8535921, 31197430, 114601065, 422891340, 1566903060, 5827192140, 21743726430, 81383916840, 305465105790, 1149489049644, 4335921660522, 16391329697528, 62091796219904, 235656705875304

Offset: 2

F. Chapoton, Apr 02 2025

Cf. A000108, A026012, A280891.

gf := ((2*x^3 + x^2 - 1)*sqrt(1 - 4*x) - 4*x^3 - 3*x^2 - 2*x + 1)/(2*x^2):
ser := series(gf, x, 30): seq(coeff(ser, x, n), n = 2..27);  # Peter Luschny, Apr 03 2025

a[n_]:=CatalanNumber[n+1]-CatalanNumber[n-1]-2CatalanNumber[n-2];Array[a,26,2] (* James C. McMahon, Apr 05 2025 *)

C = catalan_number
[C(n + 1) - C(n - 1) - 2 * C(n - 2) for n in range(2, 28)]

a(n) = [x^n] ((2*x^3 + x^2 - 1)*sqrt(1 - 4*x) - 4*x^3 - 3*x^2 - 2*x + 1)/(2*x^2). - Peter Luschny, Apr 03 2025

A382448 Triangle read by rows, defined by the two-variable g.f. (x^3y^2 + x^3y + 1)/(1 - x^2y - xy - x).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 8, 15, 8, 1, 1, 10, 29, 29, 10, 1, 1, 12, 47, 73, 47, 12, 1, 1, 14, 69, 149, 149, 69, 14, 1, 1, 16, 95, 265, 371, 265, 95, 16, 1, 1, 18, 125, 429, 785, 785, 429, 125, 18, 1, 1, 20, 159, 649, 1479, 1941, 1479, 649, 159, 20, 1

Offset: 0

F. Chapoton, Mar 26 2025

Every row is symmetric.

			Triangle begins:
  [0] [1]
  [1] [1,  1]
  [2] [1,  3,   1]
  [3] [1,  6,   6,   1]
  [4] [1,  8,  15,   8,   1]
  [5] [1, 10,  29,  29,  10,  1]
  [6] [1, 12,  47,  73,  47, 12,    1]
  [7] [1, 14,  69, 149, 149, 69,   14,   1]
  [8] [1, 16,  95, 265, 371, 265,  95,  16, 1]
  [9] [1, 18, 125, 429, 785, 785, 429, 125, 18, 1]

Similar to A008288, A103450, A382436 and A382444. Row sums are A105082.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (x^3*y^2 + x^3*y + 1)/(1 - x^2*y - x*y - x)
[list(u) for u in list(gf.O(10))]

A382439 Triangle read by rows: defined by the two-variable g.f. (x^3y^2 + x^3y - x^2y + 1) / (1 - x^2y - x*y - x).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 5, 1, 1, 7, 12, 7, 1, 1, 9, 24, 24, 9, 1, 1, 11, 40, 60, 40, 11, 1, 1, 13, 60, 124, 124, 60, 13, 1, 1, 15, 84, 224, 308, 224, 84, 15, 1, 1, 17, 112, 368, 656, 656, 368, 112, 17, 1, 1, 19, 144, 564, 1248, 1620, 1248, 564, 144, 19, 1

Offset: 0

F. Chapoton, Mar 25 2025

The alternating sum of every row n > 0 vanishes. Every row is symmetric.

			  [0] [1]
  [1] [1,  1]
  [2] [1,  2,   1]
  [3] [1,  5,   5,   1]
  [4] [1,  7,  12,   7,   1]
  [5] [1,  9,  24,  24,   9,   1]
  [6] [1, 11,  40,  60,  40,  11,   1]
  [7] [1, 13,  60, 124, 124,  60,  13,   1]
  [8] [1, 15,  84, 224, 308, 224,  84,  15,  1]
  [9] [1, 17, 112, 368, 656, 656, 368, 112, 17, 1]

Similar to A008288 and A382436. Row sums are A245990.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (x^3*y^2 + x^3*y - x^2*y + 1)/(1 - x^2*y - x*y - x)
[list(u) for u in list(gf.O(10))]

A382444 Triangle read by rows, defined by the two-variable g.f. (1 + yx^2 + (y^2 + y)x^3)/(1-(1+y)x-yx^2).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 7, 7, 1, 1, 9, 18, 9, 1, 1, 11, 34, 34, 11, 1, 1, 13, 54, 86, 54, 13, 1, 1, 15, 78, 174, 174, 78, 15, 1, 1, 17, 106, 306, 434, 306, 106, 17, 1, 1, 19, 138, 490, 914, 914, 490, 138, 19, 1, 1, 21, 174, 734, 1710, 2262, 1710, 734, 174, 21, 1

Offset: 0

F. Chapoton, Mar 25 2025

Every row is symmetric.

			Triangle begins:
  [0] [1]
  [1] [1,  1]
  [2] [1,  4,   1]
  [3] [1,  7,   7,   1]
  [4] [1,  9,  18,   9,   1]
  [5] [1, 11,  34,  34,  11,   1]
  [6] [1, 13,  54,  86,  54,  13,   1]
  [7] [1, 15,  78, 174, 174,  78,  15,   1]
  [8] [1, 17, 106, 306, 434, 306, 106,  17,  1]
  [9] [1, 19, 138, 490, 914, 914, 490, 138, 19, 1]
  ...

Similar to A008288, A103450 and A382436. Row sums are A265107.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
gf = (1 + y*x^2 + (y^2 + y)*x^3)/(1 - (1 + y)*x - y*x^2)
[list(u) for u in list(gf.O(11))]

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 9, 17, 9, 1, 1, 12, 36, 36, 12, 1, 1, 15, 64, 101, 64, 15, 1, 1, 18, 101, 227, 227, 101, 18, 1, 1, 21, 147, 440, 627, 440, 147, 21, 1, 1, 24, 202, 767, 1459, 1459, 767, 202, 24, 1, 1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1

Offset: 0

F. Chapoton, Mar 25 2025

The original definition was "Decomposition of A077938".

Every row is symmetric.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   6,    1;
  1,  9,  17,    9,    1;
  1, 12,  36,   36,   12,    1;
  1, 15,  64,  101,   64,   15,    1;
  1, 18, 101,  227,  227,  101,   18,    1;
  1, 21, 147,  440,  627,  440,  147,   21,   1;
  1, 24, 202,  767, 1459, 1459,  767,  202,  24,  1;
  1, 27, 266, 1235, 2994, 3999, 2994, 1235, 266, 27, 1;
  ...

Similar to A008288, A103450, and A382444.
Row sums are A077938.
T(2n, n) gives A339565.
Cf. A056594.

y = polygen(QQ, 'y')
x = y.parent()[['x']].gen()
inverse = 1 + (-y - 1)*x - y*x^2 + (-y^2 - y)*x^3
gf = 1 / inverse
[list(u) for u in list(gf.O(11))]

G.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).

Sum_{k=0..n} (-1)^k * T(n,k) = A056594(n). - Alois P. Heinz, Mar 25 2025

A376161 Number of support Tau-tilting modules for some algebras.

Original entry on oeis.org

3, 5, 12, 33, 98, 306, 990, 3289, 11154, 38454, 134368, 474810, 1693812, 6091780, 22064130, 80410185, 294647250, 1084922190, 4012165080, 14895504030, 55496654460, 207431394300, 777601790940, 2922867908298, 11013796950228, 41596652545756, 157434454904160, 597029454416724, 2268232385053096

Offset: 0

F. Chapoton, Sep 13 2024

nonn

See Prop. A.6 in Wang's reference for the table counting Tau-tilting modules for the linear quiver modulo the relation alpha*beta = 0.

Paolo Xausa, Table of n, a(n) for n = 0..1000
Anna Rodriguez Rasmussen, Exact Borel subalgebras of quasi-hereditary monomial algebras, arXiv:2504.01706 [math.RT], 2025. See p. 38.
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019-2022.

Cf. A005807, A007054, A000108, A016789, A023999, A329533.

a := n -> -(3*n + 2)*(-4)^(n + 1)*binomial(3/2, n + 2):
seq(a(n), n = 0..28)  # Peter Luschny, Sep 13 2024

A376161[n_] := CatalanNumber[n]*(9*n + 6)/(n + 2);
Array[A376161, 30, 0] (* Paolo Xausa, Sep 14 2024 *)

def a(n):
    return 3*(3*n+2)*binomial(2*n+4,n+2)/4/(2*n+1)/(2*n+3)

a(n) = 3*(3*n+2)*binomial(2*n+4,n+2)/(4*(2*n+1)*(2*n+3)).
a(n) = A329533(n)/(n + 1).
From Peter Luschny, Sep 13 2024: (Start)
a(n) = (3*n + 2) * [x^n] ((1 - 4*x)^(3/2) + 12*x - 2)/(4*x^2).
a(n) = A016789(n)*(3/2)*(2*n)! * [x^(2*n)] hypergeom([], [3], x^2).
a(n) = CatalanNumber(n)*(9*n + 6)/(n + 2).
a(n) = -(3*n + 2)*(-4)^(n + 1)*binomial(3/2, n + 2).
a(n) = 2^n*(9*n + 6)*(2*n - 1)!! / (n + 2)!.
a(n) = A007054(n) * (3*n + 2) / 2.
a(n) = 6*A023999(n + 1)/(n + 2)!. (End)

A371395 Triangle read by rows: T(n, k) = binomial(n + k, k) * binomial(2*n - k, n - k) / (n + 1).

Original entry on oeis.org

1, 1, 1, 2, 3, 2, 5, 10, 10, 5, 14, 35, 45, 35, 14, 42, 126, 196, 196, 126, 42, 132, 462, 840, 1008, 840, 462, 132, 429, 1716, 3564, 4950, 4950, 3564, 1716, 429, 1430, 6435, 15015, 23595, 27225, 23595, 15015, 6435, 1430

Offset: 0

F. Chapoton, Mar 21 2024

The terms can be seen as graded dimensions of a non-symmetric operad. The Koszul dual operad has Hilbert series x*(1 + x)*(1 + tx). So the current table has as Hilbert series the reverse of x*(1-x)*(1-t*x) w.r.t to x (see Sage below).

The triangle is symmetric under the exchange of k with n - k.

			Triangle begins:
  [0] [ 1],
  [1] [ 1,   1],
  [2] [ 2,   3,   2],
  [3] [ 5,  10,  10,   5],
  [4] [14,  35,  45,  35,  14],
  [5] [42, 126, 196, 196, 126, 42].

Column 0 and main diagonal are A000108.
Column 1 and subdiagonal are A001700.
Row sums are A006013.
The even bisection of the alternating row sums is A001764.
The central terms are A188681.
Cf. A085614, A250886, A371400.

T := (n, k) -> binomial(n + k, k)*binomial(2*n - k, n)/(n + 1):
seq(print(seq(T(n, k), k = 0..n)), n = 0..7);  # Peter Luschny, Mar 21 2024

T[n_, k_] := (Hypergeometric2F1[-n, -k, 1, 1] Hypergeometric2F1[-n, k - n, 1, 1]) /(n + 1); Table[T[n, k], {n, 0, 7}, {k, 0, n}] // Flatten
(* Peter Luschny, Mar 21 2024 *)

def Trow(n):
    return [binomial(n+k, k) * binomial(2*n-k, n-k) / (n+1) for k in range(n+1)]

# As the reverse of x*(1-x)*(1-t*x) w.r.t variable x.
t = polygen(QQ, 't')
x = LazyPowerSeriesRing(t.parent(), 'x').0
gf = x*(1-x)*(1-t*x)
coeffs = gf.revert() / x
for n in range(6):
    print(coeffs[n].list())

From Peter Luschny, Mar 21 2024: (Start)
T(n, k) = hypergeom([-n, -k], [1], 1)*hypergeom([-n, k - n], [1], 1)/(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(1/2)^k = A085614(n + 1).
2^n*Sum_{k=0..n} T(n, k)*(-1/2)^k = A250886(n + 1). (End)

A367872 Number of dissections of a convex (4n+4)-sided polygon into n hexagons and one square (up to equivalence).

Original entry on oeis.org

1, 4, 30, 272, 2695, 28080, 302064, 3321120, 37095201, 419276660, 4782798020, 54960207120, 635339153865, 7380876649216, 86101923008160, 1007980225327680, 11836181297108565, 139353762142502100

Offset: 0

F. Chapoton, Feb 22 2024

nonn

This sequence counts dissections of a convex 4n+4-sided polygon into one square and n hexagons, modulo a simple equivalence relation. The equivalence relation is not defined by a group, but by local moves. Consider the octagon formed by a hexagon adjacent to the square. The local move is half-rotation of such octagons.

It seems that a(n) is divisible by n+1.

			For n=0, there is just one square, so that a(0)=1. For n=1, one can dissect an octagon in 8 ways into a hexagon and a square. In this case, the equivalence relation just relates every such dissection to its half rotated image, so that a(1)=4.

Cf. A174687, A185113 (similar), A118970 (related).

Table[Binomial[5*n + 2, n]*(n + 3)/(4*n + 3), {n, 0, 50}]

for(n=0,25, print1(binomial(5*n+2,n)*(n+3)/(4*n+3), ", "))

def A367872(n):
    return binomial(5*n+2, n) * (n+3) / (4*n+3)

a(n) = binomial(5*n+2,n)*(n+3)/(4*n+3).

cp's OEIS Frontend

User: F. Chapoton

A383776 a(n) = (11*n + 3 + 6/(n+2)) * Catalan(n).

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A382880 Symmetric triangle read by rows refining A109113.

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A382668 a(n) = C(n+1) - C(n-1) - 2*C(n-2) where C(n) = A000108(n) are the Catalan numbers.

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A382448 Triangle read by rows, defined by the two-variable g.f. (x^3*y^2 + x^3*y + 1)/(1 - x^2*y - x*y - x).

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A382439 Triangle read by rows: defined by the two-variable g.f. (x^3*y^2 + x^3*y - x^2*y + 1) / (1 - x^2*y - x*y - x).

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A382444 Triangle read by rows, defined by the two-variable g.f. (1 + y*x^2 + (y^2 + y)*x^3)/(1-(1+y)*x-y*x^2).

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A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)*x - y*x^2 - (y^2 + y)*x^3).

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A376161 Number of support Tau-tilting modules for some algebras.

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Mathematica

A383776 a(n) = (11n + 3 + 6/(n+2)) Catalan(n).

A382448 Triangle read by rows, defined by the two-variable g.f. (x^3y^2 + x^3y + 1)/(1 - x^2y - xy - x).

A382439 Triangle read by rows: defined by the two-variable g.f. (x^3y^2 + x^3y - x^2y + 1) / (1 - x^2y - x*y - x).

A382444 Triangle read by rows, defined by the two-variable g.f. (1 + yx^2 + (y^2 + y)x^3)/(1-(1+y)x-yx^2).

A382436 Triangle read by rows, defined by the two-variable g.f. 1/(1 - (y + 1)x - yx^2 - (y^2 + y)*x^3).