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%I #11 Dec 04 2023 14:13:58
%S 1,4,20,130,1088,11314,141080,2052250,34118048,638102434,13260323240,
%T 303117147370,7558845354608,204203189722354,5940927689713400,
%U 185186461979970490,6157337034085736768,217523186522883467074,8136577601614291359560,321261794453042025993610,13352198666907246870560528
%N Expansion of e.g.f. exp(2*x) / (1 - 2*sinh(x)).
%H P. R. J. Asveld, <a href="https://www.fq.math.ca/Scanned/25-1/asveld.pdf">A family of Fibonacci-like sequences</a>, Fib. Quart., 25 (1987), 81-83.
%H G. Ledin, Jr., <a href="https://www.fq.math.ca/Scanned/5-1/ledin.pdf">On a certain kind of Fibonacci sums</a>, Fib. Quart., 5 (1967), 45-58. See Table IV p. 53.
%F a(n) = Sum_{k=0..n} A341725(n,k).
%F a(n) = (-1)^n*Sum_{k=0..n} (-2)^k*A341724(n,k).
%F a(n) = -1-0^n+Sum_{k=0..n} k!*Fibonacci(k+4)*Stirling2(n,k).
%F a(0) = 1; a(n) = 3^n+Sum_{k=0..n-1} (2^(n-k)-1)*binomial(n,k)*a(k).
%F a(n) ~ n! * (phi)^2 / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio.
%F a(n) = -1 + A000557(n) + A005923(n)) = - 1 + Sum_{k=0..n} |A341723(n,k) + A341724(n,k)|.
%p a := n -> -1-0^n+add(k!*combinat[fibonacci](k+4)*Stirling2(n, k), k = 0 .. n):
%p seq(a(n), n=0..20);
%p # second program:
%p a := proc(n) option remember; `if`(n=0,1,3^n+add((2^(n-k)-1)*binomial(n, k)*a(k), k=0..n-1)) end:
%p seq(a(n), n=0..20);
%p # third program:
%p a := n -> add(2^k*binomial(n, k)*add(j!*combinat[fibonacci](j+2)*Stirling2(n-k, j), j=0..n-k), k=0..n):
%p seq(a(n), n=0..20);
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*x) / (1 - 2*sinh(x)))) \\ _Michel Marcus_, Dec 04 2023
%Y Cf. A000045, A000557, A005923, A341723, A341724, A341725.
%K nonn
%O 0,2
%A _Mélika Tebni_, Dec 04 2023
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