A091144 -id:A091144 - cp's OEIS frontend

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(kn + 1, k)/(kn + 1).

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 5, 1, 0, 1, 1, 4, 12, 14, 1, 0, 1, 1, 5, 22, 55, 42, 1, 0, 1, 1, 6, 35, 140, 273, 132, 1, 0, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 0, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 0

Offset: 0

Peter Luschny, Jun 26 2022

An alternative definition is: the Fuss-Catalan sequences (A(n, k), k >= 0 ) are the main diagonals of the Fuss-Catalan triangles of order n - 1. See A355173 for the definition of a Fuss-Catalan triangle.

			Array A(n, k) begins:
[0] 1, 1, 0,   0,    0,     0,      0,       0,         0, ...  A019590
[1] 1, 1, 1,   1,    1,     1,      1,       1,         1, ...  A000012
[2] 1, 1, 2,   5,   14,    42,    132,     429,      1430, ...  A000108
[3] 1, 1, 3,  12,   55,   273,   1428,    7752,     43263, ...  A001764
[4] 1, 1, 4,  22,  140,   969,   7084,   53820,    420732, ...  A002293
[5] 1, 1, 5,  35,  285,  2530,  23751,  231880,   2330445, ...  A002294
[6] 1, 1, 6,  51,  506,  5481,  62832,  749398,   9203634, ...  A002295
[7] 1, 1, 7,  70,  819, 10472, 141778, 1997688,  28989675, ...  A002296
[8] 1, 1, 8,  92, 1240, 18278, 285384, 4638348,  77652024, ...  A007556
[9] 1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, ...  A062994

N. I. Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, vol.9 (1791), 243-251.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, (Eqs. 5.70, 7.66, and sec. 7.5, example 5).

Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338.
Jean-Luc Baril, Mireille Bousquet-Mélou, Sergey Kirgizov, and Mehdi Naima, The ascent lattice on Dyck paths, arXiv:2409.15982 [math.CO], 2024. See p. 6.
Jean-Christophe Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers, Chapter 7.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
Donald Knuth's 20th Annual Christmas Tree Lecture, (3/2)-ary Trees, Stanford Online, Video 2014.
Wojciech Młotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Math. 15:939-955, (2010).
Wikipedia, Fuss-Catalan number.

Cf. A091144 (main diagonal), A019590, A000012, A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994.
Variants: A062993, A070914.
Fuss-Catalan triangles: A123110 (order 0), A355173 (order 1), A355172 (order 2), A355174 (order 3).

A := (n, k) -> binomial(k*n + 1, k)/(k*n + 1):
for n from 0 to 9 do seq(A(n, k), k = 0..8) od;

(* See the Knuth references. In the christmas lecture Knuth has fun calculating the Fuss-Catalan development of Pi and i. *)
B[t_, n_] := Sum[Binomial[t k+1, k] z^k / (t k+1), {k, 0, n-1}] + O[z]^n
Table[CoefficientList[B[n, 9], z], {n, 0, 9}] // TableForm

A(n, k) = (1/k!) * Product_{j=1..k-1} (k*n + 1 - j).
A(n, k) = (binomial(k*n, k) + binomial(k*n, k-1)) / (k*n + 1).
Let B(t, z) = Sum_{k>=0} binomial(k*t + 1, k)*z^k / (k*t + 1), then
A(n, k) = [z^k] B(n, z).

A300474 Number of partitions of the square resulting from a sequence of n n-sections, each of which divides any part perpendicular to any of the axes.

Original entry on oeis.org

1, 1, 8, 96, 2240, 80960, 4021248, 255704064, 19878918144, 1829788646400, 194788537180160, 23556611967336448, 3191162612827078656, 478807179615908462592, 78833945248222913495040, 14133035289273287214366720, 2740751307013005651817267200

Offset: 0

Alois P. Heinz, Dec 15 2018

nonn

			a(2) = 8:
  ._______.  ._______.  ._______.  ._______.
  | | |   |  |   | | |  |_______|  |       |
  | | |   |  |   | | |  |_______|  |_______|
  | | |   |  |   | | |  |       |  |_______|
  |_|_|___|  |___|_|_|  |_______|  |_______|
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.
  .

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A091144, A236339, A237026, A300613, A322543.

a:= proc(n) option remember; `if`(n<2, 1, coeff(series(
      RootOf(G-x-2*G^n+G^(n^2), G), x, n^2-n+2), x, n^2-n+1))
    end:
seq(a(n), n=0..16);

a[0] = a[1] = 1; a[n_] := Module[{G}, G[] = 0; Do[G[x] = 2 G[x]^n - G[x]^n^2 + x + O[x]^(n^2 - n + 2) // Normal, {n^2 - n + 2}];
Coefficient[G[x], x, n^2 - n + 1]];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Dec 29 2018, after Alois P. Heinz *)

A351501 a(n) = binomial(n^2 + n - 1, n) / (n^2 + n - 1).

Original entry on oeis.org

1, 2, 15, 204, 4095, 109668, 3689595, 149846840, 7141879503, 391139588190, 24218296445200, 1673538279265020, 127715832778905150, 10670643284149377480, 968929726650218004435, 95024894699780159868144, 10011211830149283223044015

Offset: 1

F. Chapoton, May 03 2022

nonn

Empirical: In the ring of symmetric functions over the fraction field Q(q, t), let s(n) denote the Schur function indexed by n. Then (up to sign) a(n) is the coefficient of s(1^n) in nabla^(n) s(n) with q=t=1, where nabla denotes the "nabla operator" on symmetric functions.

Closely related to A177784. See also A091144.

Diagonal of A162382. Multiple of A182316.

Table[With[{c=n^2+n-1},Binomial[c,n]/c],{n,20}] (* Harvey P. Dale, Jan 01 2024 *)

from math import comb
def A351501(n): return comb(m := n**2+n-1,n)//m # Chai Wah Wu, May 07 2022

[binomial(n*n+n-1,n)/(n*n+n-1) for n in range(1,29)]

a(n) ~ c*exp(n-1/(6*n))*n^(n-5/2), where c = sqrt(e/(2*Pi)). - Stefano Spezia, May 04 2022

a(n) = n * A182316(n - 1). - F. Chapoton, Sep 22 2023

A299435 G.f.: Sum_{n>=0} binomial((n+1)^2, n)/(n+1) * x^n / (1 + x)^((n+1)^2).

Original entry on oeis.org

1, 1, 5, 51, 791, 16711, 449575, 14738537, 570860449, 25534320961, 1296145448621, 73644069770107, 4631766294581959, 319523289664700279, 23992478864877747151, 1948216141720780468561, 170121586262631029818433, 15897659114382366967974145, 1583109774987253349677203349, 167363833662976153803805436291, 18721216520653602533835176495671

Offset: 0

Paul D. Hanna, Feb 09 2018

nonn

Compare to: 1 = Sum_{n>=0} binomial(m*(n+1), n)/(n+1) * x^n / (1+x)^(m*(n+1)) holds for fixed m.

			G.f.: A(x) = 1 + x + 5*x^2 + 51*x^3 + 791*x^4 + 16711*x^5 + 449575*x^6 + 14738537*x^7 + 570860449*x^8 + 25534320961*x^9 + 1296145448621*x^10 + ...
such that
A(x) = 1/(1+x) + C(4,1)/2*x/(1+x)^4 + C(9,2)/3*x^2/(1+x)^9 + C(16,3)/4*x^3/(1+x)^16 + C(25,4)/5*x^4/(1+x)^25 + C(36,5)/6*x^5/(1+x)^36 + C(49,6)/7*x^6/(1+x)^49 + ...
more explicitly,
A(x) = 1/(1+x) + 2*x/(1+x)^4 + 12*x^2/(1+x)^9 + 140*x^3/(1+x)^16 + 2530*x^4/(1+x)^25 + 62832*x^5/(1+x)^36 + 1997688*x^6/(1+x)^49 + ... + A091144(n+1)*x^n/(1+x)^((n+1)^2) + ...

Cf. A298695, A298696, A091144.

{a(n) = my(A = sum(m=0,n,binomial((m+1)^2,m)/(m+1)*x^m/(1+x +x*O(x^n))^((m+1)^2) ) ); polcoeff(A,n)}
for(n=0,25, print1(a(n),", "))

cp's OEIS Frontend

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(k*n + 1, k)/(k*n + 1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A300474 Number of partitions of the square resulting from a sequence of n n-sections, each of which divides any part perpendicular to any of the axes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A351501 a(n) = binomial(n^2 + n - 1, n) / (n^2 + n - 1).

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Sage

Formula

A299435 G.f.: Sum_{n>=0} binomial((n+1)^2, n)/(n+1) * x^n / (1 + x)^((n+1)^2).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(kn + 1, k)/(kn + 1).