A365555 Expansion of e.g.f. 1 / (7 - 6 * exp(x))^(1/3).

1, 2, 18, 274, 5938, 167122, 5786418, 237857874, 11319677618, 612109819602, 37069480301618, 2485356833141074, 182753029186750898, 14623552941626800082, 1265002802597606144818, 117633823750542653153874, 11701922865351577653913778

Offset

0,2

Links

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

Formula

a(n) = Sum_{k=0..n} 2^k * (Product_{j=0..k-1} (3*j+1)) * Stirling2(n,k) = Sum_{k=0..n} (Product_{j=0..k-1} (6*j+2)) * Stirling2(n,k).

a(0) = 1; a(n) = Sum_{k=1..n} (6 - 4*k/n) * binomial(n,k) * a(n-k).

a(n) ~ sqrt(2*Pi) * n^(n - 1/6) / (Gamma(1/3) * 7^(1/3) * exp(n) * log(7/6)^(n + 1/3)). - Vaclav Kotesovec, Sep 09 2023

From Peter Bala, Sep 24 2023: (Start)

O.g.f. (conjectural): 1/(1 - 2*x/(1 - 7*x/(1 - 8*x/(1 - 14*x/(1 - 14*x/(1 - 21*x/(1 - ... - (6*n - 4)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction).

More generally, it appears that the o.g.f. of the sequence whose e.g.f. is 1/(r+1 - r*exp(s*x))^(m/s) corresponds to the S-fraction 1/(1 - r*m*x/(1 - s*(r+1)*x/(1 - r*(m+s)*x/(1 - 2*s(r+1)*x/(1 - r*(m+2*s)*x/(1 - 3*s(r+1)*x/( 1 - ... ))))))). This is the case r = 6, s = 1, m = 1/3. (End)

a(0) = 1; a(n) = 2*a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023

Mathematica

a[n_] := Sum[Product[6*j + 2, {j, 0, k - 1}] * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Sep 11 2023 *)
With[{nn=20}, CoefficientList[Series[1/(7-6Exp[x])^(1/3), {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Jun 21 2024 *)

Prog

(PARI) a(n) = sum(k=0, n, prod(j=0, k-1, 6*j+2)*stirling(n, k, 2));

Crossrefs

Cf. A094419, A346985, A354252, A365556, A365557.

Sequence in context: A364400 A334242 A349314 * A355472 A295183 A351768

Adjacent sequences: A365552 A365553 A365554 * A365556 A365557 A365558

Keyword

nonn

Author

Seiichi Manyama, Sep 09 2023

Status

approved