A343276 a(n) = n! * [x^n] -x*(x + 1)*exp(x)/(x - 1)^3.

0, 1, 10, 81, 652, 5545, 50886, 506905, 5480056, 64116657, 808856290, 10959016321, 158851484100, 2454385635481, 40285778016862, 700261611998985, 12853532939027056, 248482678808005345, 5047002269952482106, 107466341437781300017, 2394019421567804960380

Offset: 0

Author

Peter Luschny, Apr 20 2021

Keywords

• nonn

Crossrefs

Cf. A002775, A093964.

Programs

Maple

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

Mathematica

a[n_] := Sum[Pochhammer[n - k + 1, k]*k^2, {k, 0, n}];
Table[a[n], {n, 0, 20}]

Python

def a():
    a, b, n = 0, 1, 2
    yield 0
    while True:
        yield b
        a, b = b, -(n + 1)*a + ((2 + n*(n + 2))*b)//(n - 1)
        n += 1
A343276 = a(); print([next(A343276) for _ in range(21)])

SageMath

def a(n): return sum(rising_factorial(n - k + 1, k)*k^2 for k in (0..n))
print([a(n) for n in (0..20)])

Formula

a(n) = Sum_{k=0..n} rf(n - k + 1, k)*k^2, where rf is the rising factorial.

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

A002775(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n, k)*a(k).

a(n) = Sum_{k=1..n} k^2*k!*binomial(n,k). - Ridouane Oudra, Jun 15 2025