A346258
E.g.f.: exp(x) / (1 - 3 * x)^(1/3).
Original entry on oeis.org
1, 2, 7, 44, 421, 5366, 84907, 1601552, 35052649, 872931626, 24368595631, 753607111412, 25572085243597, 944609383245854, 37731673388579731, 1620520035001182296, 74466516342569480017, 3645540855448417250642, 189415873005295070803159, 10410429682102309433442236
Offset: 0
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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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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 *)
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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 */
A347012
E.g.f.: exp(x) / (1 - 4 * x)^(1/4).
Original entry on oeis.org
1, 2, 8, 64, 800, 13376, 278272, 6914048, 199629824, 6566164480, 242327576576, 9915111636992, 445432721932288, 21795710738038784, 1153805878313615360, 65700181140859518976, 4004182878034473254912, 260071258357260225609728, 17932703649301871611346944
Offset: 0
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g:= proc(n) option remember; `if`(n<2, 1, (4*n-3)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..18); # Alois P. Heinz, Aug 10 2021
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nmax = 18; CoefficientList[Series[Exp[x]/(1 - 4 x)^(1/4), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 4^k Pochhammer[1/4, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/4, n + 5/4, 1/4]/Sqrt[2], {n, 0, 18}]
A347014
Expansion of e.g.f.: exp(x) / (1 - 6*x)^(1/6).
Original entry on oeis.org
1, 2, 10, 116, 2140, 52856, 1627192, 59930480, 2568599056, 125553289760, 6892279877536, 419788155021632, 28090704069553600, 2048487353985408896, 161687913401407530880, 13733087614786273308416, 1248892148354210466595072, 121073054127693143488709120
Offset: 0
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g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*g(n-1)) end:
a:= n-> add(binomial(n, k)*g(k), k=0..n):
seq(a(n), n=0..17); # Alois P. Heinz, Aug 10 2021
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nmax = 17; CoefficientList[Series[Exp[x]/(1 - 6 x)^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 17}]
Table[HypergeometricU[1/6, n + 7/6, 1/6]/6^(1/6), {n, 0, 17}]
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a[n]:=if n<2 then n+1 else (6*n-4)*a[n-1]-6*(n-1)*a[n-2];
makelist(a[n],n,0,50); /* Tani Akinari, Sep 08 2023 */
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