A259688 - OEIS

%I #41 Feb 03 2025 07:10:11

%S 1,2,8,16,136,272,3968,7936,176896,353792,11184128,22368256,951878656,

%T 1903757312,104932671488,209865342976,14544442556416,29088885112832,

%U 2475749026562048,4951498053124096,507711943253426176,1015423886506852352,123460740095103991808

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

%H Alois P. Heinz, <a href="/A259688/b259688.txt">Table of n, a(n) for n = 0..483</a>

%H 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 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 a(2n) = A024283(n+1). a(2n+1)=2*a(n). - _R. J. Mathar_, Feb 03 2025

%p b:= proc(u, o) option remember;

%p       `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))

%p     end:

%p a:= n-> `if`(n::odd, b(n+2, 0), a(n+1)/2):

%p seq(a(n), n=0..24);  # _Alois P. Heinz_, Jun 15 2023

%t a[n_] := 2^Mod[n, 2]*Abs[PolyLog[-(n + 3 - Mod[n, 2]), I]];

%t Table[a[n], {n, 0, 24}] (* _Jean-François Alcover_, Dec 02 2023, after _Mikhail Kurkov_ *)

%o (PARI) a(n) = 2^(n%2)*abs(polylog(-(n + 3 - n%2), I)) \\ _Mikhail Kurkov_, Jun 13 2023

%Y Cf. A000111, A002735.

%K nonn,changed

%O 0,2

%A _N. J. A. Sloane_, Jul 04 2015

%E More terms from _Mikhail Kurkov_, Jun 13 2023

%E a(0)=1 prepended by _Alois P. Heinz_, Jun 13 2023