A360544
E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).
Original entry on oeis.org
1, 1, 7, 73, 1117, 22741, 580159, 17826985, 641494249, 26473635865, 1232945359111, 63978649829161, 3660871368065509, 229016870623703917, 15550838554432967647, 1139139301403727884521, 89544381521098908259729, 7518611017848248249471089
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-3/2*x*exp(3*x/2))/3)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((-lambertw(-3*x/2*exp(3*x/2))/(3*x/2*exp(3*x/2)))^(2/3)))
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my(N=20, x='x+O('x^N)); Vec(serlaplace((sum(k=0, N, (k+1)^(k-1)*(3*x/2*exp(3*x/2))^k/k!))^(2/3)))
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a(n) = sum(k=0, n, (3*k)^(n-k)*(3*k+2)^(k-1)*binomial(n, k))/2^(n-1);
A360548
E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).
Original entry on oeis.org
0, 1, 8, 96, 1792, 46080, 1511424, 60325888, 2837970944, 153778913280, 9432255692800, 646039266656256, 48874810528235520, 4047655951598092288, 364221261622538141696, 35384754572803304325120, 3691411033400626898796544, 411569264258973944034361344
Offset: 0
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A360548 := proc(n)
add((2*k)^(n-1)*binomial(n,k),k=1..n) ;
end proc:
seq(A360548(n),n=0..60) ; # R. J. Mathar, Mar 12 2023
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my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))
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a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));
A362653
E.g.f. satisfies A(x) = exp( x * exp(x^2) * A(x)^2 ).
Original entry on oeis.org
1, 1, 5, 55, 849, 17641, 462373, 14651295, 545025281, 23291218801, 1124589371301, 60553038168679, 3597677815336465, 233810179507710105, 16499939198003013509, 1256544674435523638671, 102713141497515307408257, 8970278754666722087785825
Offset: 0
A362774
E.g.f. satisfies A(x) = exp( x * (1+x)^2 * A(x)^2 ).
Original entry on oeis.org
1, 1, 9, 115, 2265, 59701, 1981513, 79441167, 3736418801, 201790517833, 12309193580841, 837132560820139, 62809405894333321, 5154060532188515325, 459202970647825870313, 44146740571635016905991, 4555272678073789024849377, 502153774773932684443210513
Offset: 0
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A362774 := proc(n)
n!*add((2*k+1)^(k-1) * binomial(2*k,n-k)/k!,k=0..n) ;
end proc:
seq(A362774(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-2*x*(1+x)^2)/2)))
A372279
E.g.f. A(x) satisfies A(x) = exp( x * ( exp(x) * A(x) )^(5/2) ).
Original entry on oeis.org
1, 1, 11, 181, 4461, 148101, 6202651, 314158461, 18682884681, 1276509416761, 98552772971451, 8485633118339301, 806247602665104661, 83796784405535693181, 9457590223483413296811, 1151924494605809502276301, 150602291336042725831941201
Offset: 0
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my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2/5*lambertw(-5/2*x*exp(5*x/2)))))
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a(n) = sum(k=0, n, (5*k/2)^(n-k)*(5*k/2+1)^(k-1)*binomial(n, k));
Showing 1-5 of 5 results.