A346258 E.g.f.: exp(x) / (1 - 3 * x)^(1/3).
1, 2, 7, 44, 421, 5366, 84907, 1601552, 35052649, 872931626, 24368595631, 753607111412, 25572085243597, 944609383245854, 37731673388579731, 1620520035001182296, 74466516342569480017, 3645540855448417250642, 189415873005295070803159, 10410429682102309433442236
Offset: 0
Programs
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Maple
g:= proc(n) option remember; `if`(n<2, 1, (3*n-2)*g(n-1)) end: a:= n-> add(binomial(n, k)*g(k), k=0..n): seq(a(n), n=0..19); # Alois P. Heinz, Aug 10 2021
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Mathematica
nmax = 19; CoefficientList[Series[Exp[x]/(1 - 3 x)^(1/3), {x, 0, nmax}], x] Range[0, nmax]! Table[Sum[Binomial[n, k] 3^k Pochhammer[1/3, k], {k, 0, n}], {n, 0, 19}] Table[HypergeometricU[1/3, n + 4/3, 1/3]/3^(1/3), {n, 0, 19}] With[{nn=20},CoefficientList[Series[Exp[x]/CubeRoot[1-3x],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 25 2025 *) -
Maxima
a[n]:=if n<2 then n+1 else (3*n-1)*a[n-1]+3*(1-n)*a[n-2]; makelist(a[n],n,0,50); /* Tani Akinari, Sep 08 2023 */
Formula
a(n) = Sum_{k=0..n} binomial(n,k) * A007559(k).
a(n) ~ n! * exp(1/3) * 3^n / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Aug 14 2021
a(n+2) = (3*n+5)*a(n+1)-3*(n+1)*a(n). - Tani Akinari, Sep 08 2023
Comments