A360544 - OEIS

%I #24 May 03 2023 09:10:42

%S 1,1,7,73,1117,22741,580159,17826985,641494249,26473635865,

%T 1232945359111,63978649829161,3660871368065509,229016870623703917,

%U 15550838554432967647,1139139301403727884521,89544381521098908259729,7518611017848248249471089

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

%H Seiichi Manyama, <a href="/A360544/b360544.txt">Table of n, a(n) for n = 0..352</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.

%F E.g.f.: A(x) = exp( (-2/3) * LambertW(-3*x/2 * exp(3*x/2)) ).

%F E.g.f.: A(x) = ( -LambertW(-3*x/2 * exp(3*x/2)) / (3*x/2 * exp(3*x/2)) )^(2/3).

%F E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (3*x/2 * exp(3*x/2))^k / k! )^(2/3).

%F a(n) = (1/2^(n-1)) * Sum_{k=0..n} (3*k)^(n-k) * (3*k+2)^(k-1) * binomial(n,k).

%F 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

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

%o (PARI) 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)))

%o (PARI) 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)))

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

%Y Cf. A273953, A273954, A360547.

%Y Cf. A360473, A360545.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 11 2023