OFFSET
0,3
COMMENTS
Stirling transform of A008542.
In general, for k >= 1, if e.g.f. = 1 / (k + 1 - k*exp(x))^(1/k), then a(n) ~ n! / (Gamma(1/k) * (k+1)^(1/k) * n^(1 - 1/k) * log(1 + 1/k)^(n + 1/k)). - Vaclav Kotesovec, Aug 14 2021
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..343
FORMULA
a(n) = Sum_{k=0..n} Stirling2(n,k) * A008542(k).
a(n) ~ n! / (Gamma(1/6) * 7^(1/6) * n^(5/6) * log(7/6)^(n + 1/6)). - Vaclav Kotesovec, Aug 14 2021
For n > 0, a(n) = (1/n)*Sum_{k=0..n-1} binomial(n,k)*(n+5*k)*a(k). - Tani Akinari, Aug 22 2023
O.g.f. (conjectural): 1/(1 - x/(1 - 7*x/(1 - 7*x/(1 - 14*x/(1 - 13*x/(1 - 21*x/(1 - ... - (6*n-5)*x/(1 - 7*n*x/(1 - ... ))))))))) - a continued fraction of Stieltjes-type (S-fraction). - Peter Bala, Aug 25 2023
a(0) = 1; a(n) = a(n-1) - 7*Sum_{k=1..n-1} (-1)^k * binomial(n-1,k) * a(n-k). - Seiichi Manyama, Nov 17 2023
MAPLE
g:= proc(n) option remember; `if`(n<2, 1, (6*n-5)*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/(7 - 6 Exp[x])^(1/6), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[StirlingS2[n, k] 6^k Pochhammer[1/6, k], {k, 0, n}], {n, 0, 18}]
PROG
(Maxima) a[n]:=if n=0 then 1 else (1/n)*sum(binomial(n, k)*(n+5*k)*a[k], k, 0, n-1);
makelist(a[n], n, 0, 50); /* Tani Akinari, Aug 22 2023 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Aug 09 2021
STATUS
approved