A336290
a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.
Original entry on oeis.org
1, 1, 5, 44, 628, 12994, 363548, 13141974, 593579712, 32644440048, 2141946861312, 164937634714896, 14703536203936512, 1500149281670010048, 173464224256287048576, 22541427301008492798144, 3267767649638304967827456, 525055667919614566758512640
Offset: 0
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a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 17}]
nmax = 17; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
nmax = 17; Assuming[x > 0, CoefficientList[Series[Exp[Exp[x] (EulerGamma - ExpIntegralEi[-x] + Log[x])], {x, 0, nmax}], x]] Range[0, nmax]!^2
A356927
E.g.f. satisfies A(x) = 1/(1 - x)^(A(x)/(1 - x)).
Original entry on oeis.org
1, 1, 6, 54, 676, 10980, 220488, 5289592, 147828896, 4721152320, 169723566240, 6785559484704, 298726260001728, 14362141350822720, 748845960914596608, 42092072779399215360, 2537464961261745635328, 163317885950059143238656
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (k+1)^(k-1)*(-log(1-x)/(1-x))^k/k!)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(log(1-x)/(1-x)))))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((1-x)*lambertw(log(1-x)/(1-x))/log(1-x)))
A182529
E.g.f.: exp( Sum_{n>=1} x^n * Sum_{k=1..n} 1/k^2 ) = Sum_{n>=0} a(n)*x^n/n!^2.
Original entry on oeis.org
1, 1, 7, 100, 2438, 90246, 4702142, 327233880, 29271020760, 3268118467608, 445031112068232, 72541135526581536, 13936782476047959024, 3115165518696599108976, 801181037747755210248432, 234835083029394312036638016, 77797056535321496989078410624
Offset: 0
E.g.f.: A(x) = 1 + x + 7*x^2/2!^2 + 100*x^3/3!^2 + 2438*x^4/4!^2 + 90246*x^5/5!^2 +...
such that
log(A(x)) = x + x^2*(1+1/4) + x^3*(1+1/4+1/9) + x^4*(1+1/4+1/9+1/16) + x^5*(1+1/4+1/9+1/16+1/25) + x^6*(1+1/4+1/9+1/16+1/25+1/36) +...
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{a(n)=n!^2*polcoeff(exp(sum(m=1, n+1, x^m*sum(k=1, m, 1/k^2)+x*O(x^n))), n)}
for(n=0,20,print1(a(n),", "))
A300910
Expansion of e.g.f. 1/(1 - x)^(x/(1 - x)^2).
Original entry on oeis.org
1, 0, 2, 15, 116, 1070, 11754, 149436, 2145296, 34193736, 598061160, 11377384920, 233732130312, 5153974126704, 121354505626704, 3037419444974040, 80497938647953920, 2251124265581428800, 66225476356207660224, 2044005966844402035456, 66025689709572751040640, 2227221130525199246067840, 78301158190416233445985920
Offset: 0
1/(1 - x)^(x/(1 - x)^2) = 1 + 2*x^2/2! + 15*x^3/3! + 116*x^4/4! + 1070*x^5/5! + 11754*x^6/6! + 149436*x^7/7! + ...
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a:=series(1/(1-x)^(x/(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[1/(1 - x)^(x/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k k! (HarmonicNumber[k] - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
A308535
Expansion of e.g.f. 1/(1 - x)^log(1 + x) (even powers only).
Original entry on oeis.org
1, 2, 22, 608, 31764, 2695992, 338441112, 58961602464, 13614906576528, 4024831155397536, 1482492491866434912, 665729215100873644800, 358022910151079384324928, 227174478580352888344068480, 167941710127005880795828894080, 143087068385495604780364250426880
Offset: 0
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nmax = 15; Table[(CoefficientList[Series[1/(1 - x)^Log[1 + x], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A347978
E.g.f.: 1/(1 + x)^(1/(1 - x)).
Original entry on oeis.org
1, -1, 0, -3, 4, -30, 186, -630, 11600, -26712, 1005480, -2581920, 117196872, -485308824, 17734457664, -131070696120, 3387342915840, -43890398953920, 801577841697216, -17363169328243392, 233460174245351040, -7968629225100337920, 84363134551361043840
Offset: 0
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nmax = 22; CoefficientList[Series[1/(1 + x)^(1/(1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
A024167[n_] := n! Sum[(-1)^(k + 1)/k, {k, 1, n}]; a[0] = 1; a[n_] := a[n] = -Sum[Binomial[n - 1, k - 1] A024167[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]
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my(x='x+O('x^30)); Vec(serlaplace(1/(1+x)^(1/(1-x)))) \\ Michel Marcus, Sep 22 2021
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