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A308521 Expansion of e.g.f. (sec(x) + tan(x))/(1 - 2*x).

%I A308521 #16 Jun 11 2022 11:44:18
%S A308521 1,3,13,80,645,6466,77653,1087414,17400009,313208098,6264212481,
%T A308521 137813028374,3307515383741,85995422345522,2407872025035597,
%U A308521 72236162654825222,2311557224345919249,78592945837626597442,2829346052559437183353,107515150026347498080246
%N A308521 Expansion of e.g.f. (sec(x) + tan(x))/(1 - 2*x).
%C A308521 Boustrophedon transform of A000165 (double factorial of even numbers).
%H A308521 Robert Israel, <a href="/A308521/b308521.txt">Table of n, a(n) for n = 0..403</a>
%H A308521 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A308521 a(n) ~ n! * (sec(1/2) + tan(1/2)) * 2^n. - _Vaclav Kotesovec_, Jun 07 2019
%p A308521 N:= 25: # for a(0)..a(N)
%p A308521 S:= series((sec(x)+tan(x))/(1-2*x),x,N+1):
%p A308521 seq(coeff(S,x,n)*n!,n=0..N); # _Robert Israel_, Jun 06 2019
%t A308521 nmax = 19; CoefficientList[Series[(Sec[x] + Tan[x])/(1 - 2 x), {x, 0, nmax}], x] Range[0, nmax]!
%t A308521 t[n_, 0] := 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, 20, 0]
%o A308521 (Python)
%o A308521 from itertools import count, islice, accumulate
%o A308521 def A308521_gen(): # generator of terms
%o A308521     blist, m = tuple(), 1
%o A308521     for i in count(1):
%o A308521         yield (blist := tuple(accumulate(reversed(blist),initial=m)))[-1]
%o A308521         m *= 2*i
%o A308521 A308521_list = list(islice(A308521_gen(),30)) # _Chai Wah Wu_, Jun 11 2022
%Y A308521 Cf. A000111, A000165, A230960, A231200, A296792.
%K A308521 nonn
%O A308521 0,2
%A A308521 _Ilya Gutkovskiy_, Jun 04 2019