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A347014
Expansion of e.g.f.: exp(x) / (1 - 6*x)^(1/6).
3
1, 2, 10, 116, 2140, 52856, 1627192, 59930480, 2568599056, 125553289760, 6892279877536, 419788155021632, 28090704069553600, 2048487353985408896, 161687913401407530880, 13733087614786273308416, 1248892148354210466595072, 121073054127693143488709120
OFFSET
0,2
COMMENTS
Binomial transform of A008542.
In general, for k >= 1, if e.g.f. = exp(x) / (1 - k*x)^(1/k), then a(n) ~ n! * exp(1/k) * k^n / (Gamma(1/k) * n^(1 - 1/k)). - Vaclav Kotesovec, Aug 14 2021
FORMULA
a(n) = Sum_{k=0..n} binomial(n,k) * A008542(k).
a(n) ~ n! * exp(1/6) * 6^n / (Gamma(1/6) * n^(5/6)). - Vaclav Kotesovec, Aug 14 2021
a(n+2) = (6*n+8)*a(n+1) - 6*(n+1)*a(n). - Tani Akinari, Sep 08 2023
MAPLE
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
MATHEMATICA
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}]
PROG
(Maxima) 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 */
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 10 2021
STATUS
approved