A380648
Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-4*x)/(1 + x)^4 ).
Original entry on oeis.org
1, 8, 188, 7816, 475096, 38289504, 3857806144, 467330651456, 66209818738176, 10747317030192640, 1967261819870112256, 400989528160028255232, 90087157573721153554432, 22119056538323287540637696, 5893098619063477612068864000, 1693364632974231188010697990144
Offset: 0
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nmax=16; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-4*x]/(1 + x)^4, {x, 0, nmax}]], x]Range[0, nmax-1]! (* Stefano Spezia, Feb 06 2025 *)
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a(n) = 4*n!*sum(k=0, n, (4*n+4)^(k-1)*binomial(4*n+4, n-k)/k!);
A380808
Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x) / (1 + x*exp(-x)) ).
Original entry on oeis.org
1, 3, 24, 335, 6812, 183397, 6168406, 249350285, 11785793352, 638146503593, 38960123581154, 2648475653518081, 198429466488527164, 16246940820392924189, 1443430758561178861758, 138305198841617791230533, 14217431594874334746229520, 1560842183273111251153540945
Offset: 0
A380762
Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-x * (1 + x)^2) / (1 + x) ).
Original entry on oeis.org
1, 2, 15, 208, 4249, 115656, 3946879, 162225680, 7807264497, 430828353280, 26825288214031, 1860715287986688, 142304071119852745, 11897080341213068288, 1079508321205459768575, 105660694801273960216576, 11097101798773200862180321, 1244852059489783737208012800
Offset: 0
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a(n, q=1, r=1, s=1, t=2, u=1) = q*n!*sum(k=0, n, (r*n+(s-r)*k+q)^(k-1)*binomial(r*u*n+((s-r)*u+t)*k+q*u, n-k)/k!);
A088694
E.g.f: A(x) = f(x*A(x)^3), where f(x) = (1+4*x)*exp(x).
Original entry on oeis.org
1, 5, 159, 10228, 1009253, 135069696, 22882888555, 4696799559488, 1133128780421385, 314294095403352064, 98550149514670698071, 34473870245560804316160, 13310522831484403851847981, 5622806397207798234900070400, 2579680348909056700728913816227
Offset: 0
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Table[n!*SeriesCoefficient[((1+4*x)*E^x)^(3*n+1)/(3*n+1),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 24 2014 *)
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a(n)=n!*polcoeff(((1+4*x)*exp(x))^(3*n+1)+x*O(x^n),n,x)/(3*n+1)
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