A306836 Expansion of e.g.f. (sec(x) + tan(x))*exp(x)*BesselI(1,2*x)/x.

1, 2, 5, 15, 50, 187, 779, 3626, 18830, 108737, 694156, 4865168, 37175028, 307674135, 2742199399, 26185972811, 266727515558, 2886665421697, 33078737906112, 400112950565602, 5094396150412628, 68107059089489867, 953882607872927724, 13966992152276252633, 213399920723596392052

Offset: 0

Author

Ilya Gutkovskiy, Apr 16 2019

Keywords

• nonn

Comments

Boustrophedon transform of Motzkin numbers (A001006).

Links

• Index entries for sequences related to boustrophedon transform

Crossrefs

Cf. A000111, A000753, A001006.

Programs

Mathematica

nmax = 24; CoefficientList[Series[(Sec[x] + Tan[x]) Exp[x] BesselI[1, 2 x]/x, {x, 0, nmax}], x] Range[0, nmax]!
t[n_, 0] := Hypergeometric2F1[(1 - n)/2, -n/2, 2, 4]; t[n_, k_] := t[n, k] = t[n, k - 1] + t[n - 1, n - k]; a[n_] := t[n, n]; Array[a, 25, 0]

Python

from itertools import accumulate, count, islice
def A306836_gen(): # generator of terms
    blist, a, b = (1,), 1, 1
    yield from blist
    for i in count(2):
        yield (blist := tuple(accumulate(reversed(blist),initial=b)))[-1]
        a, b = b, (b*(2*i+1)+(3*i-3)*a)//(i+2)
A306836_list = list(islice(A306836_gen(),40)) # Chai Wah Wu, Jun 12 2022