A380258 Expansion of e.g.f. exp( (1/(1-5*x)^(2/5) - 1)/2 ).

1, 1, 8, 106, 1954, 46082, 1323064, 44750644, 1741897340, 76672512316, 3764746706176, 203976645319448, 12086590557877144, 777464693554778776, 53948773488864143072, 4016672567726156437744, 319379204127841984947472, 27010128651142535536409360, 2420802590890201251989984128

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Eric Weisstein's World of Mathematics, Bell Polynomial.

Formula

a(n) = Sum_{k=0..n} 5^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 5^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

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

From Vaclav Kotesovec, Feb 04 2026: (Start)

a(n) = 35*(n-3)*a(n-1) - 105*(5*n^2 - 35*n + 63)*a(n-2) + 875*(n-4)*(5*n^2 - 40*n + 84)*a(n-3) - 7*(3125*n^4 - 56250*n^3 + 383125*n^2 - 1170000*n + 1351287)*a(n-4) + (65625*n^5 - 1640625*n^4 + 16471875*n^3 - 83015625*n^2 + 210003885*n - 213300674)*a(n-5) - 175*(n-6)*(n-5)*(625*n^4 - 13750*n^3 + 113750*n^2 - 419375*n + 581361)*a(n-6) + 125*(n-7)*(n-6)*(n-5)*(5*n - 33)*(5*n - 31)*(5*n - 29)*(5*n - 27)*a(n-7).

a(n) ~ 5^(n + 1/7) * n^(n - 5/14) / (sqrt(7) * exp(n + 1/2 - 7*5^(-5/7)*n^(2/7)/2)) * (1 + 1/(2*5^(10/7)*n^(3/7))). (End)

Mathematica

CoefficientList[Series[Exp[ (1/(1-5*x)^(2/5) - 1)/2 ], {x, 0, 18}], x]Range[0, 18]! (* Stefano Spezia, Mar 31 2025 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(1-5*x)^(2/5)-1)/2)))

Crossrefs

Cf. A049376, A375173, A380257.

Cf. A004211, A025168.

Sequence in context: A129278 A369512 A112701 * A395122 A392734 A099695

Adjacent sequences: A380255 A380256 A380257 * A380259 A380260 A380261

Keyword

nonn

Author

Seiichi Manyama, Jan 18 2025

Status

approved