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A306822
Expansion of e.g.f. (sec(x) + tan(x))*exp(2*x)*BesselI(0,2*x).
1
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
0,2
COMMENTS
Boustrophedon transform of central binomial coefficients (A000984).
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
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]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Apr 16 2019
STATUS
approved