A352070 Expansion of e.g.f. 1/(1 - log(1 + 3*x))^(1/3).

1, 1, 1, 10, 10, 604, -1844, 107344, -1201400, 42193576, -875584376, 29853569008, -880141783184, 32865860907424, -1216481572723616, 51296026356128512, -2244334822166729600, 106984479644794783360, -5358207684820194270080, 286466413246622566048000

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

a(n) = Sum_{k=0..n} 3^(n-k) * (Product_{j=0..k-1} (3*j+1)) * Stirling1(n,k).

For n > 0, a(n) = n!*Sum_{k=1..n} a(n-k)*(2/n/3-1/k)*(-3)^k/(n-k)!. - Tani Akinari, Sep 07 2023

a(n) ~ -(-1)^n * 3^(n-1) * n! / (n * log(n)^(4/3)) * (1 - 4*(1+gamma)/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 07 2023

Mathematica

m = 19; Range[0, m]! * CoefficientList[Series[(1 - Log[1 + 3*x])^(-1/3), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *)
Table[Sum[3^(n-k) * Product[3*j+1, {j, 0, k-1}] * StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2023 *)

Prog

(PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-log(1+3*x))^(1/3)))
(PARI) a(n) = sum(k=0, n, 3^(n-k)*prod(j=0, k-1, 3*j+1)*stirling(n, k, 1));
(Maxima) a[n]:=if n=0 then 1 else n!*sum(a[n-k]*(2/n/3-1/k)*(-3)^k/(n-k)!, k, 1, n);
makelist(a[n], n, 0, 50); /* Tani Akinari, Sep 07 2023 */

Crossrefs

Cf. A006252, A097397, A352073.

Cf. A007559, A133480, A352113, A352118.

Sequence in context: A065243 A105750 A220449 * A318968 A288051 A288504

Adjacent sequences: A352067 A352068 A352069 * A352071 A352072 A352073

Keyword

sign

Author

Seiichi Manyama, Mar 05 2022

Status

approved