A372279 E.g.f. A(x) satisfies A(x) = exp( x * ( exp(x) * A(x) )^(5/2) ).

1, 1, 11, 181, 4461, 148101, 6202651, 314158461, 18682884681, 1276509416761, 98552772971451, 8485633118339301, 806247602665104661, 83796784405535693181, 9457590223483413296811, 1151924494605809502276301, 150602291336042725831941201

Offset

0,3

Links

Table of n, a(n) for n=0..16.

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

Formula

E.g.f.: A(x) = exp( -2/5 * LambertW(-5*x/2 * exp(5*x/2)) ).

E.g.f.: A(x) = ( -LambertW(-5*x/2 * exp(5*x/2)) / (5*x/2 * exp(5*x/2)) )^(2/5).

E.g.f.: A(x) = ( Sum_{k>=0} (k+1)^(k-1) * (5*x/2 * exp(5*x/2))^k / k! )^(2/5).

a(n) = Sum_{k=0..n} (5*k/2)^(n-k) * (5*k/2+1)^(k-1) * binomial(n,k).

a(n) ~ sqrt(1 + LambertW(exp(-1))) * 5^(n-1) * n^(n-1) / (exp(n - 2/5) * 2^(n-1) * LambertW(exp(-1))^n). - Vaclav Kotesovec, May 06 2024

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-2/5*lambertw(-5/2*x*exp(5*x/2)))))
(PARI) a(n) = sum(k=0, n, (5*k/2)^(n-k)*(5*k/2+1)^(k-1)*binomial(n, k));

Crossrefs

Cf. A273953, A273954, A360544, A360547.

Cf. A372278.

Sequence in context: A321848 A112943 A258381 * A057618 A270663 A068648

Adjacent sequences: A372276 A372277 A372278 * A372280 A372281 A372282

Keyword

nonn

Author

Seiichi Manyama, Apr 25 2024

Status

approved