A365569 Expansion of e.g.f. 1 / (6 - 5 * exp(x))^(3/5).

1, 3, 27, 387, 7659, 193491, 5948091, 215446563, 8984708235, 423944899443, 22328393101659, 1298429924941251, 82625791930962219, 5711012035686681363, 426058604580805219323, 34121803137713388036963, 2919847869159667841599947, 265868538017899566748612275

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = Sum_{k=0..n} (Product_{j=0..k-1} (5*j+3)) * Stirling2(n,k).

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

a(n) ~ sqrt(2*Pi) * n^(n + 1/10) / (6^(3/5) * Gamma(3/5) * exp(n) * log(6/5)^(n + 3/5)). - Vaclav Kotesovec, Nov 11 2023

a(0) = 1; a(n) = 3*a(n-1) - 6*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023

Mathematica

a[n_] := Sum[Product[5*j + 3, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20}, CoefficientList[Series[1/(6-5*Exp[x])^(3/5), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Nov 03 2024 *)

Prog

(PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 5*j+3)*stirling(n, k, 2));

Crossrefs

Cf. A094418, A346984, A365568, A365570.

Sequence in context: A365794 A138436 A141057 * A365586 A201696 A011781

Adjacent sequences: A365566 A365567 A365568 * A365570 A365571 A365572

Keyword

nonn

Author

Seiichi Manyama, Sep 09 2023

Status

approved