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A337444 Expansion of e.g.f. (1 + 2*x) * exp(x) / (sec(x) + tan(x)).

%I A337444 #9 Jun 11 2022 14:03:23
%S A337444 1,2,0,-1,-8,-5,-50,79,-696,3375,-22830,156719,-1205244,9952735,
%T A337444 -88770370,847488719,-8633147152,93430030975,-1070638898070,
%U A337444 12950169210479,-164886839860420,2204373939190495,-30873660197736330,452059981437456399,-6906967824497129048
%N A337444 Expansion of e.g.f. (1 + 2*x) * exp(x) / (sec(x) + tan(x)).
%C A337444 Inverse boustrophedon transform of odd numbers.
%H A337444 <a href="/index/Bo#boustrophedon">Index entries for sequences related to boustrophedon transform</a>
%F A337444 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * (2*k+1) * A000111(n-k).
%t A337444 nmax = 24; CoefficientList[Series[(1 + 2 x) Exp[x]/(Sec[x] + Tan[x]), {x, 0, nmax}], x] Range[0, nmax]!
%t A337444 t[n_, 0] := 2 n + 1; t[n_, k_] := t[n, k] = t[n, k - 1] - t[n - 1, n - k]; a[n_] := t[n, n]; Table[a[n], {n, 0, 24}]
%o A337444 (Python)
%o A337444 from itertools import count, islice, accumulate
%o A337444 from operator import sub
%o A337444 def A337444_gen(): # generator of terms
%o A337444     blist = tuple()
%o A337444     for i in count(1,2):
%o A337444         yield (blist := tuple(accumulate(reversed(blist),func=sub,initial=i)))[-1]
%o A337444 A337444_list = list(islice(A337444_gen(),30)) # _Chai Wah Wu_, Jun 11 2022
%Y A337444 Cf. A000111, A000754, A005408.
%K A337444 sign
%O A337444 0,2
%A A337444 _Ilya Gutkovskiy_, Aug 27 2020