A343582 a(n) = (-1)^n*n!*[x^n] exp(-3*x)/(1 - 2*x).

1, 1, 5, -3, 105, -807, 10413, -143595, 2304081, -41453775, 829134549, -18240782931, 437779321785, -11382260772087, 318703306401405, -9561099177693243, 305955173729230497, -10402475906664696735, 374489132640316502949, -14230587040330864850595, 569223481613238080808201

Offset: 0

Author

Peter Luschny, Apr 24 2021

Keywords

• sign

Comments

The row polynomials of the rencontres numbers (A008290) evaluated at -1/2 and normalized by (-2)^n.

Crossrefs

Cf. A008290, A000166, A000354.

Programs

Maple

egf := exp(-3*x)/(1 - 2*x): ser := series(egf, x, 32):
seq((-1)^n*n!*coeff(ser, x, n), n=0..20);

Mathematica

a[n_] := (-2)^n Sum[Binomial[n, k] Subfactorial[n - k] (-2)^(-k), {k, 0, n}];
Table[a[n], {n, 0, 20}]

Python

def A343582():
    a, b, n = 1, 5, 3
    yield 1
    yield a
    while True:
        yield b
        a, b = b, 6*(n - 1)*a - (2*n - 3)*b
        n += 1
a = A343582(); print([next(a) for _ in range(21)])

Formula

a(n) = (-2)^n*Sum_{k=0..n} binomial(n, k)*subfactorial(n - k)*(-1/2)^k.

a(n) = 6*(n - 1)*a(n - 2) - (2*n - 3)*a(n - 1) for n >= 3.