A337445 E.g.f.: 1 / ((sec(x) + tan(x)) * (1 - x)).

1, 0, 1, 1, 9, 29, 235, 1373, 12369, 103385, 1084371, 11574289, 141594233, 1818356773, 25656355803, 382941579733, 6146456787873, 104279900050865, 1879443080591011, 35680329646116377, 713976964110565065, 14988564748268742269, 329817773336305467819

Offset: 0

Author

Ilya Gutkovskiy, Aug 27 2020

Keywords

• nonn

Comments

Inverse boustrophedon transform of factorials.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..450
• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000111, A000142, A230960, A230961.

Programs

Maple

b:= proc(u, o) option remember;
     `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember; add((-1)^(n-k)
      *binomial(n, k)*k!*b(n-k, 0), k=0..n)
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Aug 15 2021

Mathematica

nmax = 22; CoefficientList[Series[1/((Sec[x] + Tan[x]) (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := n!; t[n_, k_] := t[n, k] = t[n, k - 1] - t[n - 1, n - k]; a[n_] := t[n, n]; Table[a[n], {n, 0, 22}]

Python

from itertools import count, islice, accumulate
from operator import sub
def A337445_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=m)))[-1]
        m *= i
A337445_list = list(islice(A337445_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * k! * A000111(n-k).

a(n) ~ n! * cos(1) / (1 + sin(1)). - Vaclav Kotesovec, Aug 31 2020