A370985
Expansion of e.g.f. (1/x) * Series_Reversion( x*(1 - x^3*exp(x)) ).
Original entry on oeis.org
1, 0, 0, 6, 24, 60, 3000, 45570, 403536, 10644984, 297562320, 5517833310, 142801022760, 5076208052916, 150282366476424, 4713707747551530, 189345734667052320, 7517503455423740400, 295622259241028433696, 13370535071068474177974, 642403497550155241197240
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(serreverse(x*(1-x^3*exp(x)))/x))
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a(n) = sum(k=0, n\3, k^(n-3*k)*(n+k)!/(k!*(n-3*k)!))/(n+1);
A371042
E.g.f. satisfies A(x) = 1 + x^2*exp(x*A(x)).
Original entry on oeis.org
1, 0, 2, 6, 12, 140, 1470, 10122, 114296, 1874952, 25462170, 379431470, 7546461252, 151797222876, 3066316693622, 72101615826450, 1843378516587120, 47860832586054032, 1338908395558366386, 40675047500003794902, 1282380661224172506620
Offset: 0
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nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^3]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Mar 10 2024 *)
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a(n) = n!*sum(k=0, n\2, k^(n-2*k)*binomial(n-2*k+1, k)/((n-2*k+1)*(n-2*k)!));
A371043
E.g.f. satisfies A(x) = 1 + x^2*A(x)*exp(x*A(x)).
Original entry on oeis.org
1, 0, 2, 6, 36, 380, 3630, 47082, 725816, 12132360, 235801530, 5083309550, 119757623172, 3103443520476, 87082536196838, 2632399338834930, 85471932351187440, 2961803643600574352, 109154615479427298546, 4264407640037365789014, 175984871341042826680700
Offset: 0
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