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

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..352

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

Formula

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

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2*lambertw(-3/2*x*exp(3*x/2))/3)))
(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)))
(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)))
(PARI) a(n) = sum(k=0, n, (3*k)^(n-k)*(3*k+2)^(k-1)*binomial(n, k))/2^(n-1);

Crossrefs

Cf. A273953, A273954, A360547.

Cf. A360473, A360545.

Sequence in context: A352118 A258379 A325930 * A352123 A364938 A134281

Adjacent sequences: A360541 A360542 A360543 * A360545 A360546 A360547

Keyword

nonn

Author

Seiichi Manyama, Feb 11 2023

Status

approved