A360545 -id:A360545 - cp's OEIS frontend

A360544 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).

1, 1, 7, 73, 1117, 22741, 580159, 17826985, 641494249, 26473635865, 1232945359111, 63978649829161, 3660871368065509, 229016870623703917, 15550838554432967647, 1139139301403727884521, 89544381521098908259729, 7518611017848248249471089

Offset: 0

Seiichi Manyama, Feb 11 2023

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..352
Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A273953, A273954, A360547.

Cf. A360473, A360545.

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-3/2*x*exp(3*x/2))/3)))

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)))

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)))

a(n) = sum(k=0, n, (3*k)^(n-k)*(3*k+2)^(k-1)*binomial(n, k))/2^(n-1);

E.g.f.: A(x) = exp( (-2/3) * LambertW(-3*x/2 * exp(3*x/2)) ).
E.g.f.: A(x) = ( -LambertW(-3*x/2 * exp(3*x/2)) / (3*x/2 * exp(3*x/2)) )^(2/3).
E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (3*x/2 * exp(3*x/2))^k / k! )^(2/3).
a(n) = (1/2^(n-1)) * Sum_{k=0..n} (3*k)^(n-k) * (3*k+2)^(k-1) * binomial(n,k).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^(n-1) * exp(n - 2/3) * LambertW(exp(-1))^n). - Vaclav Kotesovec, Feb 17 2023

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

Seiichi Manyama, Feb 11 2023

nonn

Eric Weisstein's World of Mathematics, Lambert W-Function.

Cf. A100526, A216857, A360545.

Cf. A038049, A201595, A214225, A360547.

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

my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-lambertw(-2*x*exp(2*x))/2)))

a(n) = sum(k=1, n, (2*k)^(n-1)*binomial(n, k));

E.g.f.: A(x) = (-1/2) * LambertW(-2*x * exp(2*x)).
a(n) = Sum_{k=1..n} (2*k)^(n-1) * binomial(n,k) = 4^(n-1) * A100526(n).
a(n) ~ sqrt(1 + LambertW(exp(-1))) * 2^(n-1) * n^(n-1) / (LambertW(exp(-1))^n * exp(n)). - Vaclav Kotesovec, Feb 17 2023

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A360544 E.g.f. satisfies A(x) = exp( x * ( exp(x) * A(x) )^(3/2) ).

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A360548 E.g.f. satisfies A(x) = x * exp( 2*(x + A(x)) ).

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