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}]
A347013
E.g.f.: exp(x) / (1 - 5 * x)^(1/5).
Original entry on oeis.org
1, 2, 9, 88, 1361, 28182, 726889, 22414988, 803913441, 32867765002, 1508608850249, 76804271962848, 4294870015118641, 261673684619584862, 17252970318529474089, 1223896705010751194068, 92946073511938131386561, 7523666291578066678172562, 646658551118777059833155209
Offset: 0
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g:= proc(n) option remember; `if`(n<2, 1, (5*n-4)*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 - 5 x)^(1/5), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[Binomial[n, k] 5^k Pochhammer[1/5, k], {k, 0, n}], {n, 0, 18}]
Table[HypergeometricU[1/5, n + 6/5, 1/5]/5^(1/5), {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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