A305923
Expansion of e.g.f. exp(x)/(1 - log(1 + x) - log(1 + x)^2).
Original entry on oeis.org
1, 2, 6, 21, 105, 580, 4332, 33173, 333057, 3249334, 41175698, 485901669, 7470988137, 102962077608, 1870375878472, 29342124588357, 617978798588225, 10818920340476010, 260570216908845406, 5009431835664474101, 136578252867673635369, 2844357524328057280332, 87134882338620095240484
Offset: 0
E.g.f.: A(x) = 1 + 2*x/1! + 6*x^2/2! + 21*x^3/3! + 105*x^4/4! + 580*x^5/5! + 4332*x^6/6! + ...
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a:=series(exp(x)/(1-log(1+x)-log(1+x)^2),x=0,23): seq(n!*coeff(a,x,n),n=0..22); # Paolo P. Lava, Mar 26 2019
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nmax = 22; CoefficientList[Series[Exp[x]/(1 - Log[1 + x] - Log[1 + x]^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Sum[Binomial[n, k] StirlingS1[k, j] j! Fibonacci[j + 1], {j, 0, k}], {k, 0, n}], {n, 0, 22}]
A364822
Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).
Original entry on oeis.org
1, 2, 9, 56, 465, 4832, 60249, 876416, 14570145, 272502272, 5662834089, 129446475776, 3228012339825, 87205172928512, 2537079010567929, 79084060649947136, 2629496833837277505, 92893490657046167552, 3474733464040954877769, 137195165161622584426496, 5702069567580948171751185
Offset: 0
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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
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a[n_]:=n!*SeriesCoefficient[Cosh[x]/(1 - 2*Sinh[x]),{x,0,n}]; Array[a,21,0] (* Stefano Spezia, Nov 07 2023 *)
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my(x='x+O('x^30)); Vec(serlaplace(cosh(x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Nov 07 2023
A367887
Expansion of e.g.f. exp(2*x) / (1 - 2*sinh(x)).
Original entry on oeis.org
1, 4, 20, 130, 1088, 11314, 141080, 2052250, 34118048, 638102434, 13260323240, 303117147370, 7558845354608, 204203189722354, 5940927689713400, 185186461979970490, 6157337034085736768, 217523186522883467074, 8136577601614291359560, 321261794453042025993610, 13352198666907246870560528
Offset: 0
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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);
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my(x='x+O('x^30)); Vec(serlaplace(exp(2*x) / (1 - 2*sinh(x)))) \\ Michel Marcus, Dec 04 2023
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