A340474
a(n) = n! [x^n] LW(T(x)), where T(x) = -W(-x) Euler's tree function, W(x) is the Lambert W function, and LW(x) = W(-W(x))/(-W(x)) (A340473).
Original entry on oeis.org
1, 1, 3, 22, 209, 2756, 43717, 839686, 18581425, 470707192, 13352676101, 420875581754, 14566375690297, 549877190829604, 22472783629465093, 989043215802778966, 46631075599107558113, 2345376059569552767344, 125350843842721213505029, 7095169059445749303612946
Offset: 0
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W := x -> LambertW(x): T := x -> -W(-x): LW := x -> W(-W(x))/(-W(x)):
ser := series(LW(T(x)), x, 24): seq(n!*coeff(ser, x, n), n=0..19);
A356926
E.g.f. satisfies A(x)^A(x) = 1/(1 - x)^exp(x).
Original entry on oeis.org
1, 1, 2, 3, 10, 35, 121, 1092, 5216, 39321, 558643, 2433508, 48144944, 688652549, 2176310995, 145742587616, 1334993574032, 5551320939809, 799648465754835, 1049695714507276, 90069170433616208, 6281942689646504501, -53282051261767839293, 2356158301117802408472
Offset: 0
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nmax = 23; A[_] = 1;
Do[A[x_] = ((1 - x)^(-Exp[x]))^(1/A[x]) + O[x]^(nmax+1) // Normal, {nmax}];
CoefficientList[A[x], x]*Range[0, nmax]! (* Jean-François Alcover, Mar 04 2024 *)
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my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, (-k+1)^(k-1)*(-exp(x)*log(1-x))^k/k!)))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(lambertw(-exp(x)*log(1-x)))))
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my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x)*log(1-x)/lambertw(-exp(x)*log(1-x))))
A227866
Derived from von Mangoldt matrix sequence.
Original entry on oeis.org
1, 1, 4, 27, 64, 3125, 288, 823543, 147456, 4251528, 460800, 285311670611, 111974400, 302875106592253, 3251404800, 13436928000, 106542032486400, 827240261886336764177, 1053455155200000, 1978419655660313589123979, 102395841085440000
Offset: 0
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Clear[nn, t, n, k, i, s]; nn = 20; t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] := t[n, k] = If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[t[k - i, n], {i, 1, n - 1}]]; Exp[Table[Limit[Zeta[s]*Sum[If[n == 1, 0, t[n, k]]/k^(s - 1), {k, 1, n}], s -> 1], {n, 0, nn}]]*(Range[nn + 1] - 1)!
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