A306822 Expansion of e.g.f. (sec(x) + tan(x))*exp(2*x)*BesselI(0,2*x).

1, 3, 11, 46, 207, 988, 4989, 26734, 152827, 937212, 6192741, 44191654, 340575513, 2829201638, 25252605283, 241269232186, 2457951274627, 26602476272908, 304845053785469, 3687342610303174, 46948693772419597, 627657623728640182, 8790742273959180703, 128716280124796774354

Offset: 0

Author

Ilya Gutkovskiy, Apr 16 2019

Keywords

• nonn

Comments

Boustrophedon transform of central binomial coefficients (A000984).

Links

• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000111, A000753, A000984.

Programs

Mathematica

nmax = 23; CoefficientList[Series[(Sec[x] + Tan[x]) Exp[2 x] BesselI[0, 2 x], {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := Binomial[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, 24, 0]

Python

from itertools import count, islice, accumulate
def A306822_gen(): # generator of terms
    blist, m = tuple(), 1
    for i in count(1):
        yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
        m = m*(4*i-2)//i
A306822_list = list(islice(A306822_gen(),20)) # Chai Wah Wu, Jun 11 2022

Formula

a(n) ~ BesselI(0, Pi) * 2^(n + 5/2) * n^(n + 1/2) / (Pi^(n + 1/2) * exp(n - Pi)). - Vaclav Kotesovec, May 04 2024