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

1, 2, 8, 16, 136, 272, 3968, 7936, 176896, 353792, 11184128, 22368256, 951878656, 1903757312, 104932671488, 209865342976, 14544442556416, 29088885112832, 2475749026562048, 4951498053124096, 507711943253426176, 1015423886506852352, 123460740095103991808

Offset: 0

Author

N. J. A. Sloane, Jul 04 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..483
• C. Krishnamachary and M. Bheemasena Rao, Determinants whose elements are Eulerian, prepared Bernoullian and other numbers, J. Indian Math. Soc., 14 (1922), 55-62, 122-138 and 143-146. See p. 146. [Annotated scanned copy]

Crossrefs

Cf. A000111, A002735.

Programs

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= n-> `if`(n::odd, b(n+2, 0), a(n+1)/2):
seq(a(n), n=0..24);  # Alois P. Heinz, Jun 15 2023

Mathematica

a[n_] := 2^Mod[n, 2]*Abs[PolyLog[-(n + 3 - Mod[n, 2]), I]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 02 2023, after Mikhail Kurkov *)

PARI

a(n) = 2^(n%2)*abs(polylog(-(n + 3 - n%2), I)) \\ Mikhail Kurkov, Jun 13 2023

Formula

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

a(2n) = A024283(n+1). a(2n+1)=2*a(n). - R. J. Mathar, Feb 03 2025

Extensions

More terms from Mikhail Kurkov, Jun 13 2023

a(0)=1 prepended by Alois P. Heinz, Jun 13 2023