A307593 -id:A307593 - cp's OEIS frontend

A307594 Expansion of e.g.f. (sec(x) + tan(x))*exp(-x)/(1 - x).

1, 1, 2, 7, 28, 145, 880, 6221, 50048, 452097, 4531440, 49919461, 599595192, 7799387921, 109232872424, 1638888843541, 26226252579296, 445889920626817, 8026518496420896, 152509898418037765, 3050274960962524520, 64056803495548131665, 1409264093116606297080

Offset: 0

Ilya Gutkovskiy, Apr 17 2019

nonn

Boustrophedon transform of A000166.

Index entries for sequences related to boustrophedon transform

Cf. A000111, A000166, A230960, A307593.

nmax = 22; CoefficientList[Series[(Sec[x] + Tan[x]) Exp[-x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := (-1)^n HypergeometricPFQ[{-n, 1}, {}, 1]; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 23, 0]

from itertools import accumulate, count, islice
def A307594_gen(): # generator of terms
    blist, a, b = tuple(), 1, -1
    for n in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=a)))[-1]
        a, b = a*n+b, -b
A307594_list = list(islice(A307594_gen(),30)) # Chai Wah Wu, Jun 11 2022

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

Original entry on oeis.org

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

Ilya Gutkovskiy, Aug 15 2021

nonn

Inverse boustrophedon transform of A000522.

Binomial transform of A337445.

Index entries for sequences related to boustrophedon transform

Cf. A000111, A000522, A307593, A337445.

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}]

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

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

cp's OEIS Frontend

A307594 Expansion of e.g.f. (sec(x) + tan(x))*exp(-x)/(1 - x).

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