A364822 Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).

1, 2, 9, 56, 465, 4832, 60249, 876416, 14570145, 272502272, 5662834089, 129446475776, 3228012339825, 87205172928512, 2537079010567929, 79084060649947136, 2629496833837277505, 92893490657046167552, 3474733464040954877769, 137195165161622584426496, 5702069567580948171751185

Offset: 0

Author

Mélika Tebni, Nov 07 2023

Keywords

• nonn

Comments

Conjectures: For p prime (p > 2), a(p) == 2 (mod p).

For n = 2^m (m natural number), a(n) == 1 (mod n).

Links

• Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, Sums related to the Fibonacci sequence, Husson University (2021). See Table 2 p. 5.

Crossrefs

Cf. A000556, A000557, A005923, A332257, A341724.

Programs

Maple

a := n -> add(binomial(n,2*k)*add(j!*combinat[fibonacci](j+2)*Stirling2(n-2*k,j), j=0..n-2*k), k=0..floor(n/2)):
seq(a(n), n = 0 .. 20);
# second program:
b := proc(n) option remember; `if`(n = 0, 1, 2+2*add(binomial(n,2*k-1)*b(n-2*k+1), k=1..floor((n+1)/2))) end:
a := proc(n) `if`(n = 0, 1, b(n)/2) end: seq(a(n), n = 0 .. 20);
# third program:
(1/2)*((exp(-x) + exp(x))/(1 + exp(-x) - exp(x))): series(%, x, 21):
seq(n!*coeff(%, x, n), n = 0..20);  # Peter Luschny, Nov 07 2023

Mathematica

a[n_]:=n!*SeriesCoefficient[Cosh[x]/(1 - 2*Sinh[x]),{x,0,n}]; Array[a,21,0] (* Stefano Spezia, Nov 07 2023 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace(cosh(x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Nov 07 2023

Formula

a(n) = A000556(n) + A332257(n), for n > 0.

a(n) = (-1)^n*Sum_{k=0..floor(n/2)} A341724(n,2*k).

a(n) = (A000556(n) + A005923(n)) / 2.

a(n) ~ n! / (2*log((1 + sqrt(5))/2)^(n+1)).