A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).

1, 1, 6, 56, 706, 11186, 213156, 4742256, 120571676, 3447128796, 109427729096, 3818008773536, 145196289453656, 5976489668054296, 264685744187399536, 12548508890339297856, 634022724191046592016, 34007862777419093053456, 1929842567333195106456416

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} 3^(n-k) * |Stirling1(n,k)| * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * |Stirling1(n,k)| * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.

a(n) = (1/exp(1/2)) * (-3)^n * n! * Sum_{k>=0} binomial(-2*k/3,n)/(2^k * k!).

From Vaclav Kotesovec, Feb 04 2026: (Start)

a(n) = 15*(n-2)*a(n-1) - 5*(18*n^2 - 90*n + 115)*a(n-2) + (270*n^3 - 2430*n^2 + 7335*n - 7424)*a(n-3) - 135*(n-4)*(n-3)*(3*n^2 - 21*n + 37)*a(n-4) + 27*(n-5)*(n-4)*(n-3)*(3*n - 13)*(3*n - 11)*a(n-5).

a(n) ~ 3^(n + 1/5) * n^(n - 3/10) / (sqrt(5) * exp(n - 5*n^(2/5)/(2*3^(3/5)) + 1/2)) * (1 + 1/(2*3^(6/5)*n^(1/5))). (End)

Mathematica

CoefficientList[Series[Exp[ (1/(1-3*x)^(2/3) - 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/(1-3*x)^(2/3)-1)/2)))

Crossrefs

Cf. A049376, A375173, A380258.

Cf. A004211, A380214.

Sequence in context: A385806 A215507 A112699 * A386880 A093197 A303921

Adjacent sequences: A380254 A380255 A380256 * A380258 A380259 A380260

Keyword

nonn

Author

Seiichi Manyama, Jan 18 2025

Status

approved