A162382 -id:A162382 - cp's OEIS frontend

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

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

A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 14, 0, 1, 5, 26, 91, 143, 42, 0, 1, 6, 40, 204, 612, 728, 132, 0, 1, 7, 57, 385, 1771, 4389, 3876, 429, 0, 1, 8, 77, 650, 4095, 16380, 32890, 21318, 1430, 0, 1, 9, 100, 1015, 8184, 46376, 158224, 254475, 120175, 4862, 0

Offset: 0

Seiichi Manyama, Jul 26 2025

			Square array begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  0,   2,    7,    15,     26,     40,      57, ...
  0,   5,   30,    91,    204,    385,     650, ...
  0,  14,  143,   612,   1771,   4095,    8184, ...
  0,  42,  728,  4389,  16380,  46376,  109668, ...
  0, 132, 3876, 32890, 158224, 548340, 1533939, ...

Wikipedia, Fuss-Catalan number

Columns k=0..10 give A000007, A000108, A006013, A006632, A118971, A130564(n+1), A130565(n+1), A234466, A234513, A234573, A235340.
Main diagonal gives A177784(n+1).
Cf. A162382.

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);

For k > 0, A(n,k) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=k+1 and r=k.

G.f. of column k: (1/x) Series_Reverion( x*(1-x)^k ).

cp's OEIS Frontend

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

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