A346982 Expansion of e.g.f. 1 / (4 - 3 * exp(x))^(1/3).
1, 1, 5, 41, 477, 7201, 133685, 2945881, 75145677, 2177900241, 70687244965, 2539879312521, 100086803174077, 4291845333310081, 198954892070938645, 9914294755149067961, 528504758009562261677, 30010032597449931644721, 1808359960001658961070725
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..373
Crossrefs
Programs
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Maple
g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end: b:= proc(n, m) option remember; `if`(n=0, g(m), m*b(n-1, m)+b(n-1, m+1)) end: a:= n-> b(n, 0): seq(a(n), n=0..18); # Alois P. Heinz, Aug 09 2021 -
Mathematica
nmax = 18; CoefficientList[Series[1/(4 - 3 Exp[x])^(1/3), {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[StirlingS2[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 18}]
Formula
a(n) = Sum_{k=0..n} Stirling2(n,k) * A007559(k).
a(n) ~ n! / (Gamma(1/3) * 2^(2/3) * n^(2/3) * log(4/3)^(n + 1/3)). - Vaclav Kotesovec, Aug 14 2021
From Peter Bala, Aug 22 2023: (Start)
O.g.f. (conjectural): 1/(1 - x/(1 - 4*x/(1 - 4*x/(1 - 8*x/(1 - 7*x/(1 - 12*x/(1 - ... - (3*n-2)*x/(1 - 4*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 equal to 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 = 3, s = 1, m = 1/3. (End)
a(0) = 1; a(n) = Sum_{k=1..n} (3 - 2*k/n) * binomial(n,k) * a(n-k). - Seiichi Manyama, Sep 09 2023
a(0) = 1; a(n) = a(n-1) - 4*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 16 2023
Comments