A303943 Expansion of 1/(1 - x/(1 - 1^2*x/(1 - 2^2*x/(1 - 3^2*x/(1 - 4^2*x/(1 - ...)))))), a continued fraction.

1, 1, 2, 8, 76, 1540, 53684, 2812148, 205054036, 19805016628, 2444724910292, 375282530128052, 70102075181928148, 15655136160745164340, 4118456236678107528404, 1260512820941791994429876, 444069171743010266366969044, 178408825363590577961830752052

Offset

0,3

Comments

Invert transform of Euler (or secant) numbers (A000364), shifted right one place.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..243

N. J. A. Sloane, Transforms

Formula

a(n) ~ 2^(4*n - 1) * n^(2*n - 3/2) / (Pi^(2*n - 3/2) * exp(2*n)). - Vaclav Kotesovec, Jun 08 2019

Maple

b:= proc(u, o) option remember;
      `if`(u+o=0, 1, add(b(o-1+j, u-j), j=1..u))
    end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*b((i-1)*2, 0), i=1..n))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 13 2018
# Alternative:
T := proc(n, k) option remember; if k = 0 then 1 else if k = n then T(n, k-1)
else (n - k)^2 * T(n, k - 1) + T(n - 1, k) fi fi end:
a := n -> T(n, n): seq(a(n), n = 0..17);  # Peter Luschny, Oct 02 2023

Mathematica

nmax = 17; CoefficientList[Series[1/(1 - x/(1 + ContinuedFractionK[-k^2 x, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 17; CoefficientList[Series[1/(1 - x Sum[Abs[EulerE[2 k]] x^k, {k, 0, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[Abs[EulerE[2 (k - 1)]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

Crossrefs

Cf. A000290, A000364, A112934, A305532.

Sequence in context: A204552 A002668 A193205 * A295345 A329968 A093062

Adjacent sequences: A303940 A303941 A303942 * A303944 A303945 A303946

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 04 2018

Status

approved