A296792 -id:A296792 - cp's OEIS frontend

A308521 Expansion of e.g.f. (sec(x) + tan(x))/(1 - 2*x).

1, 3, 13, 80, 645, 6466, 77653, 1087414, 17400009, 313208098, 6264212481, 137813028374, 3307515383741, 85995422345522, 2407872025035597, 72236162654825222, 2311557224345919249, 78592945837626597442, 2829346052559437183353, 107515150026347498080246

Offset: 0

Ilya Gutkovskiy, Jun 04 2019

nonn

Boustrophedon transform of A000165 (double factorial of even numbers).

Robert Israel, Table of n, a(n) for n = 0..403
Index entries for sequences related to boustrophedon transform

Cf. A000111, A000165, A230960, A231200, A296792.

N:= 25: # for a(0)..a(N)
S:= series((sec(x)+tan(x))/(1-2*x),x,N+1):
seq(coeff(S,x,n)*n!,n=0..N); # Robert Israel, Jun 06 2019

nmax = 19; CoefficientList[Series[(Sec[x] + Tan[x])/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := 2^n n!; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 20, 0]

from itertools import count, islice, accumulate
def A308521_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m *= 2*i
A308521_list = list(islice(A308521_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) ~ n! * (sec(1/2) + tan(1/2)) * 2^n. - Vaclav Kotesovec, Jun 07 2019

A308681 E.g.f.: (sec(x) - tan(x)) / sqrt(1 - 2*x).

Original entry on oeis.org

1, 0, 2, 7, 60, 519, 5890, 76637, 1158808, 19770383, 377036646, 7939301349, 183033429524, 4584731740471, 123994410402122, 3601004174824573, 111771076844177328, 3692510526181175583, 129364120799128910158, 4790645026641043053269, 186981399898552187792620

Offset: 0

Ilya Gutkovskiy, Aug 15 2021

nonn

Inverse boustrophedon transform of A001147.

Index entries for sequences related to boustrophedon transform

Cf. A000111, A001147, A296792, A337445.

nmax = 20; CoefficientList[Series[(Sec[x] - Tan[x])/Sqrt[1 - 2 x], {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := (2 n - 1)!!; 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, 20}]

from itertools import count, islice, accumulate
from operator import sub
def A308681_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 *= (2*i-1)
A308681_list = list(islice(A308681_gen(),30)) # Chai Wah Wu, Jun 11 2022

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A001147(k) * A000111(n-k).

a(n) ~ (1 - sin(1/2)) * 2^(n + 1/2) * n^n / (cos(1/2) * exp(n)). - Vaclav Kotesovec, Aug 23 2021

cp's OEIS Frontend

A308521 Expansion of e.g.f. (sec(x) + tan(x))/(1 - 2*x).

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