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

%I A306836 #28 Jun 12 2022 09:38:48
%S A306836 1,2,5,15,50,187,779,3626,18830,108737,694156,4865168,37175028,
%T A306836 307674135,2742199399,26185972811,266727515558,2886665421697,
%U A306836 33078737906112,400112950565602,5094396150412628,68107059089489867,953882607872927724,13966992152276252633,213399920723596392052
%N A306836 Expansion of e.g.f. (sec(x) + tan(x))*exp(x)*BesselI(1,2*x)/x.
%C A306836 Boustrophedon transform of Motzkin numbers (A001006).
%H A306836 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%t A306836 nmax = 24; CoefficientList[Series[(Sec[x] + Tan[x]) Exp[x] BesselI[1, 2 x]/x, {x, 0, nmax}], x] Range[0, nmax]!
%t A306836 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 A306836 (Python)
%o A306836 from itertools import accumulate, count, islice
%o A306836 def A306836_gen(): # generator of terms
%o A306836     blist, a, b = (1,), 1, 1
%o A306836     yield from blist
%o A306836     for i in count(2):
%o A306836         yield (blist := tuple(accumulate(reversed(blist),initial=b)))[-1]
%o A306836         a, b = b, (b*(2*i+1)+(3*i-3)*a)//(i+2)
%o A306836 A306836_list = list(islice(A306836_gen(),40)) # _Chai Wah Wu_, Jun 12 2022
%Y A306836 Cf. A000111, A000753, A001006.
%K A306836 nonn
%O A306836 0,2
%A A306836 _Ilya Gutkovskiy_, Apr 16 2019