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%I #23 Nov 08 2023 16:44:12
%S 1,2,9,56,465,4832,60249,876416,14570145,272502272,5662834089,
%T 129446475776,3228012339825,87205172928512,2537079010567929,
%U 79084060649947136,2629496833837277505,92893490657046167552,3474733464040954877769,137195165161622584426496,5702069567580948171751185
%N Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).
%C Conjectures: For p prime (p > 2), a(p) == 2 (mod p).
%C For n = 2^m (m natural number), a(n) == 1 (mod n).
%H Paul Kinlaw, Michael Morris, and Samanthak Thiagarajan, <a href="https://www.researchgate.net/publication/350886459_SUMS_RELATED_TO_THE_FIBONACCI_SEQUENCE">Sums related to the Fibonacci sequence</a>, Husson University (2021). See Table 2 p. 5.
%F a(n) = A000556(n) + A332257(n), for n > 0.
%F a(n) = (-1)^n*Sum_{k=0..floor(n/2)} A341724(n,2*k).
%F a(n) = (A000556(n) + A005923(n)) / 2.
%F a(n) ~ n! / (2*log((1 + sqrt(5))/2)^(n+1)).
%p 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)):
%p seq(a(n), n = 0 .. 20);
%p # second program:
%p 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:
%p a := proc(n) `if`(n = 0, 1, b(n)/2) end: seq(a(n), n = 0 .. 20);
%p # third program:
%p (1/2)*((exp(-x) + exp(x))/(1 + exp(-x) - exp(x))): series(%, x, 21):
%p seq(n!*coeff(%, x, n), n = 0..20); # _Peter Luschny_, Nov 07 2023
%t a[n_]:=n!*SeriesCoefficient[Cosh[x]/(1 - 2*Sinh[x]),{x,0,n}]; Array[a,21,0] (* _Stefano Spezia_, Nov 07 2023 *)
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(cosh(x) / (1 - 2*sinh(x)))) \\ _Michel Marcus_, Nov 07 2023
%Y Cf. A000556, A000557, A005923, A332257, A341724.
%K nonn
%O 0,2
%A _Mélika Tebni_, Nov 07 2023