A394763 Main diagonal of A394582.

1, 2, 14, 120, 2002, 31668, 846260, 19669408, 725520510, 22302380100, 1050565700340, 40208894434800, 2306233390653636, 105690931811261384, 7147427113615588360, 381647055369434628224, 29733096067155146595510, 1812925835184119075505540, 159895014723794744296802900, 10961834110187084182910698000

Offset

1,2

Links

Table of n, a(n) for n=1..20.

Andrii Husiev, Extended Central Factorial Numbers and the Flickering Operator, arXiv:2605.06689 [math.GM], 2026. See references.

Formula

a(n) = A394582(n, n).

a(2*n-1) = A008957(3n-2,2n-1).

a(2*n) = 2 * n * A008957(3n-1,2n).

a(n) = (1/(2*n-1)!) * Sum_{i=0..2*n-1} (-1)^(i+1) * binomial(2*n-1, i) * (i-n+1)^(3*n-2). [Simplified by Vaclav Kotesovec, May 15 2026]

a(n) = Sum_{j=0..3*n-2} binomial(3*n-2, j) * (1-n)^(3*n-2-j) * Stirling2(j, 2*n-1).

a(n) = A395021(3*n-2, 2*n-1).

a(n) = ((3*n-2)! / (2*n-1)!) * [x^(3*n-2)] ( (2*sinh(x/2))^(2*n-1) * exp(x/2) ).

a(n) ~ (1 - (-1)^n*r/(2+r)) * exp(2*n) * r^(r*n) * (1+r)^(3*n-1) * n^(n - 3/2) / (sqrt(2*Pi*(2 - 2*r - r^2)) * (2+r)^((2+r)*n - 1)), where r = 0.1647414545521878292908344008181647954486720209245... is the root of the equation r*exp(3/(1+r)) = 2+r. - Vaclav Kotesovec, May 16 2026

Mathematica

Table[1/(2*n-1)! * Sum[(-1)^(i+1)*Binomial[2*n-1, i]*(i-n+1)^(3*n-2), {i, 0, 2*n-1}], {n, 1, 20}] (* Vaclav Kotesovec, May 15 2026 *)

Prog

(Python)
def a394582_diagonal(k_limit):
    T = [[1] * (k_limit + 1) for _ in range(k_limit + 1)]
    for k in range(2, k_limit + 1):
        for n in range(2, k_limit + 1):
            if k % 2 == 0:
                T[n][k] = n * T[n][k-1]
            else:
                T[n][k] = n * T[n][k-1] + T[n-1][k]
    return [T[i][i] for i in range(1, k_limit + 1)]
print(a394582_diagonal(10))
(Python)
def aList(n: int) -> list[int]:
    A = [1] * (n + 1 )
    for i in range(2, n + 1):
        for k in range(2, n + 1):
            A[k] = i * A[k-1] + (k % 2) * A[k]
    return A[1:]
def a(n): return aList(n)[-1]
L = [a(n) for n in range(1, 21)]; print(L)  # Peter Luschny, Apr 04 2026

Crossrefs

Cf. A394582, A395021, A036969, A008957.

Sequence in context: A390889 A192457 A351856 * A155728 A267906 A199560

Adjacent sequences: A394760 A394761 A394762 * A394764 A394765 A394766

Keyword

nonn

Author

Husiev Andrii Alekseevich, Mar 31 2026

Status

approved