A306836 - OEIS

%I #28 Jun 12 2022 09:38:48

%S 1,2,5,15,50,187,779,3626,18830,108737,694156,4865168,37175028,

%T 307674135,2742199399,26185972811,266727515558,2886665421697,

%U 33078737906112,400112950565602,5094396150412628,68107059089489867,953882607872927724,13966992152276252633,213399920723596392052

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

%C Boustrophedon transform of Motzkin numbers (A001006).

%H <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>

%t nmax = 24; CoefficientList[Series[(Sec[x] + Tan[x]) Exp[x] BesselI[1, 2 x]/x, {x, 0, nmax}], x] Range[0, nmax]!

%t 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]

%o (Python)

%o from itertools import accumulate, count, islice

%o def A306836_gen(): # generator of terms

%o     blist, a, b = (1,), 1, 1

%o     yield from blist

%o     for i in count(2):

%o         yield (blist := tuple(accumulate(reversed(blist),initial=b)))[-1]

%o         a, b = b, (b*(2*i+1)+(3*i-3)*a)//(i+2)

%o A306836_list = list(islice(A306836_gen(),40)) # _Chai Wah Wu_, Jun 12 2022

%Y Cf. A000111, A000753, A001006.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Apr 16 2019