A347071 E.g.f.: exp(x) * (sec(x) - tan(x)) / (1 - x).

1, 1, 2, 5, 20, 95, 580, 3999, 32272, 288783, 2898300, 31807679, 382253808, 4964649079, 69546528636, 1042802172359, 16688865840384, 283667092507743, 5106507590277564, 97017597229232975, 1940428937186428720, 40747978365579886375, 896469940257304900700

Offset: 0

Author

Ilya Gutkovskiy, Aug 15 2021

Keywords

• nonn

Comments

Inverse boustrophedon transform of A000522.

Binomial transform of A337445.

Links

• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000111, A000522, A307593, A337445.

Programs

Mathematica

nmax = 22; CoefficientList[Series[Exp[x] (Sec[x] - Tan[x])/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := n! Sum[1/k!, {k, 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 A347071_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 = m*i + 1
A347071_list = list(islice(A347071_gen(),30)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000522(k) * A000111(n-k).

a(n) = Sum_{k=0..n} binomial(n,k) * A337445(k).

a(n) ~ n! * exp(1)*(1 - sin(1))/cos(1). - Vaclav Kotesovec, Aug 23 2021