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A259688 Related to Euler numbers, expansion of e.g.f. tan(x)^2.

%I A259688 #41 Feb 03 2025 07:10:11
%S A259688 1,2,8,16,136,272,3968,7936,176896,353792,11184128,22368256,951878656,
%T A259688 1903757312,104932671488,209865342976,14544442556416,29088885112832,
%U A259688 2475749026562048,4951498053124096,507711943253426176,1015423886506852352,123460740095103991808
%N A259688 Related to Euler numbers, expansion of e.g.f. tan(x)^2.
%H A259688 Alois P. Heinz, <a href="/A259688/b259688.txt">Table of n, a(n) for n = 0..483</a>
%H A259688 C. Krishnamachary and M. Bheemasena Rao, <a href="/A000108/a000108_10.pdf">Determinants whose elements are Eulerian, prepared Bernoullian and other numbers</a>, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. See p. 146. [Annotated scanned copy]
%F A259688 a(2n+1) = b(2n+3) for n >= 0, a(2n) = b(2n+3)/2 for n >= 0 where b(n) = A000111(n). - _Mikhail Kurkov_, Jun 13 2023
%F A259688 a(2n) = A024283(n+1). a(2n+1)=2*a(n). - _R. J. Mathar_, Feb 03 2025
%p A259688 b:= proc(u, o) option remember;
%p A259688       `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
%p A259688     end:
%p A259688 a:= n-> `if`(n::odd, b(n+2, 0), a(n+1)/2):
%p A259688 seq(a(n), n=0..24);  # _Alois P. Heinz_, Jun 15 2023
%t A259688 a[n_] := 2^Mod[n, 2]*Abs[PolyLog[-(n + 3 - Mod[n, 2]), I]];
%t A259688 Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Dec 02 2023, after _Mikhail Kurkov_ *)
%o A259688 (PARI) a(n) = 2^(n%2)*abs(polylog(-(n + 3 - n%2), I)) \\ _Mikhail Kurkov_, Jun 13 2023
%Y A259688 Cf. A000111, A002735.
%K A259688 nonn
%O A259688 0,2
%A A259688 _N. J. A. Sloane_, Jul 04 2015
%E A259688 More terms from _Mikhail Kurkov_, Jun 13 2023
%E A259688 a(0)=1 prepended by _Alois P. Heinz_, Jun 13 2023