A367887 Expansion of e.g.f. exp(2*x) / (1 - 2*sinh(x)).

1, 4, 20, 130, 1088, 11314, 141080, 2052250, 34118048, 638102434, 13260323240, 303117147370, 7558845354608, 204203189722354, 5940927689713400, 185186461979970490, 6157337034085736768, 217523186522883467074, 8136577601614291359560, 321261794453042025993610, 13352198666907246870560528

Offset: 0

Author

Mélika Tebni, Dec 04 2023

Keywords

• nonn

Links

• P. R. J. Asveld, A family of Fibonacci-like sequences, Fib. Quart., 25 (1987), 81-83.
• G. Ledin, Jr., On a certain kind of Fibonacci sums, Fib. Quart., 5 (1967), 45-58. See Table IV p. 53.

Crossrefs

Cf. A000045, A000557, A005923, A341723, A341724, A341725.

Programs

Maple

a := n -> -1-0^n+add(k!*combinat[fibonacci](k+4)*Stirling2(n, k), k = 0 .. n):
seq(a(n), n=0..20);
# second program:
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:
seq(a(n), n=0..20);
# third program:
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):
seq(a(n), n=0..20);

PARI

my(x='x+O('x^30)); Vec(serlaplace(exp(2*x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Dec 04 2023

Formula

a(n) = Sum_{k=0..n} A341725(n,k).

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

a(n) = -1-0^n+Sum_{k=0..n} k!*Fibonacci(k+4)*Stirling2(n,k).

a(0) = 1; a(n) = 3^n+Sum_{k=0..n-1} (2^(n-k)-1)*binomial(n,k)*a(k).

a(n) ~ n! * (phi)^2 / (sqrt(5) * (log(phi))^(n+1)), where phi is the golden ratio.

a(n) = -1 + A000557(n) + A005923(n) = -1 + Sum_{k=0..n} |A341723(n,k) + A341724(n,k)|.