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A260786 Twice the Euler or up/down numbers A000111.

%I A260786 #16 Apr 17 2023 11:18:12
%S A260786 2,2,2,4,10,32,122,544,2770,15872,101042,707584,5405530,44736512,
%T A260786 398721962,3807514624,38783024290,419730685952,4809759350882,
%U A260786 58177770225664,740742376475050,9902996106248192,138697748786275802,2030847773013704704,31029068327114173810,493842960380415967232
%N A260786 Twice the Euler or up/down numbers A000111.
%H A260786 S. T. Thompson, <a href="/A260786/a260786.pdf">Problem E754: Skew Ordered Sequences</a>, Amer. Math. Monthly, 54 (1947), 416-417. [Annotated scanned copy]
%F A260786 a(0)=a(1)=2; thereafter a(n) = (1/4)*Sum_{k=1..n} binomial(n-1, k-1)*a(k-1)*a(n-k).
%p A260786 f:=proc(n) option remember;
%p A260786 if n <= 1 then 2 else (1/4)*add(binomial(n-1,k-1)*f(k-1)*f(n-k),k=1..n); fi;
%p A260786 end;
%p A260786 [seq(f(n),n=0..30)];
%o A260786 (Python)
%o A260786 from itertools import accumulate, islice
%o A260786 def A260786_gen(): # generator of terms
%o A260786     yield from (2,2)
%o A260786     blist = (0,2)
%o A260786     while True:
%o A260786         yield (blist := tuple(accumulate(reversed(blist),initial=0)))[-1]
%o A260786 A260786_list = list(islice(A260786_gen(),30)) # _Chai Wah Wu_, Apr 17 2023
%Y A260786 Cf. A000111.
%Y A260786 Apart from initial terms, same as A001250.
%K A260786 nonn
%O A260786 0,1
%A A260786 _N. J. A. Sloane_, Aug 04 2015