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

Author

Ilya Gutkovskiy, Jun 04 2019

Keywords

• nonn

Comments

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

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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]

Python

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

Formula

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