A354018
Expansion of e.g.f. -log(1-x)/(1 + log(1-x) - log(1-x)^2).
Original entry on oeis.org
0, 1, 3, 20, 172, 1864, 24248, 368136, 6388128, 124711944, 2705241672, 64550432352, 1680280323984, 47383464508080, 1438986494794704, 46821994627363968, 1625069178022566528, 59927028756823323648, 2339899614887520358656, 96439023491479275172608
Offset: 0
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Table[Sum[k! * Fibonacci[k] * Abs[StirlingS1[n,k]], {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, May 15 2022 *)
With[{nn=20},CoefficientList[Series[-Log[1-x]/(1+Log[1-x]-Log[1-x]^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Nov 22 2024 *)
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my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-x)/(1+log(1-x)-log(1-x)^2))))
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a(n) = sum(k=0, n, k!*fibonacci(k)*abs(stirling(n, k, 1)));
A358031
Expansion of e.g.f. (1 - log(1-x))/(1 + log(1-x) * (1 - log(1-x))).
Original entry on oeis.org
1, 2, 8, 52, 450, 4878, 63474, 963744, 16724016, 326497632, 7082393136, 168995017200, 4399028766192, 124051494462816, 3767315220903072, 122581568808533760, 4254486275273419008, 156890997080103149568, 6125936704495619486976, 252480641031903073955328
Offset: 0
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f:= proc(n) local k; add(k!*combinat:-fibonacci(k+2)*abs(Stirling1(n,k)),k=0..n) end proc:
map(f, [$0..30]); # Robert Israel, Oct 25 2022
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With[{nn=20},CoefficientList[Series[(1-Log[1-x])/(1+Log[1-x](1-Log[1-x])),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Jan 25 2024 *)
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1-log(1-x))/(1+log(1-x)*(1-log(1-x)))))
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a(n) = sum(k=0, n, k!*fibonacci(k+2)*abs(stirling(n, k, 1)));
A354015
Expansion of e.g.f. 1/(1 - x)^(1 - log(1-x)).
Original entry on oeis.org
1, 1, 4, 18, 106, 750, 6188, 58184, 613156, 7149780, 91319712, 1267089912, 18969355656, 304646227704, 5222700792528, 95169251327040, 1836450816902928, 37403582826055824, 801728489886598848, 18037821249349491360, 424970923585819603872, 10462258547232790348512
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^(1-log(1-x))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-log(1-x)*(1-log(1-x)))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, (1+2*sum(k=1, j-1, 1/k))*v[i-j+1]/(i-j)!)); v;
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a(n) = sum(k=0, n, k!*sum(j=0, k\2, 1/(j!*(k-2*j)!))*abs(stirling(n, k, 1)));
Showing 1-3 of 3 results.