A217144 Alternating sums of squares of Bell numbers (A000110).

1, 0, 4, 21, 204, 2500, 38709, 730420, 16409180, 430786429, 13019414196, 447437830704, 17306961847705, 746907935199264, 35695643204860420, 1876878693983656605, 107956500727342113004, 6758630146906528885412, 458470139353155531447869

Offset: 0

Author

Emanuele Munarini, Sep 27 2012

Keywords

• nonn

Crossrefs

Cf. A000110, A005001, A087650, A217143.

Programs

Maxima

makelist(sum((-1)^(n-k)*belln(k)^2,k,0,n),n,0,30);

Python

from itertools import accumulate, islice
def A217144_gen(): # generator of terms
    yield 1
    blist, b, c, f = (1,), 1, 1, 1
    while True:
        blist = list(accumulate(blist, initial=(b:=blist[-1])))
        yield (f:=-f)*(c := c+f*b**2)
A217144_list = list(islice(A217144_gen(),20)) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k)*Bell(k)^2.