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

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

Offset: 0

Author

Ilya Gutkovskiy, Aug 15 2021

Keywords

• nonn

Comments

Inverse boustrophedon transform of A001147.

Links

• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000111, A001147, A296792, A337445.

Programs

Mathematica

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}]

Python

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

Formula

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