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A194006
E.g.f.: exp( exp(3*x) - 2*exp(2*x) + 2*exp(x) - 1 ).
2
1, 1, 4, 23, 149, 1122, 9895, 99017, 1095022, 13225701, 173261337, 2444494252, 36889309869, 592107098533, 10063638652228, 180435656257819, 3401248197894641, 67204507885768562, 1388196723844580331, 29907194746593479677, 670590559444043372630
OFFSET
0,3
LINKS
FORMULA
a(n) = Sum_{k=1..n} Sum_{j=k..n} Sum_{i=0..floor((j-k)/3)} j!*(-1)^i*binomial(j-3*i-1,k-1)*binomial(k,k-i)*Stirling2(n,j)/k!, n>1, a(0)=1.
a(n) ~ n^(n + 1/2) * exp(-1 + 2*exp(r) - 2*exp(2*r) + exp(3*r) - n) / (r^n * sqrt(exp(r) * r * (2*(1 + r) - 4 * exp(r) * (1 + 2*r) + exp(2*r) * (3 + 9*r)))), where r = LambertW(n)/3 + (4*n^(1/3)/LambertW(n)^(1/3) - 2) / (9*n^(2/3)/LambertW(n)^(2/3) + 9*n^(2/3)/LambertW(n)^(5/3) - 4*n^(1/3)*(3 + 2*LambertW(n))/LambertW(n)^(4/3) + 6/LambertW(n) + 2). - Vaclav Kotesovec, Jul 05 2022
MAPLE
E:=exp( exp(3*x) - 2*exp(2*x) + 2*exp(x) - 1 ):
S:= series(E, x, 51):
seq(coeff(S, x, n)*n!, n=0..50); # Robert Israel, Oct 05 2020
PROG
(Maxima) a(n):=if n=0 then 1 else sum(sum(j!*(sum((-1)^i*binomial(j-3*i-1, k-1)* binomial(k, k-i), i, 0, (j-k)/3))*stirling2(n, j), j, k, n)/k!, k, 1, n);
CROSSREFS
Sequence in context: A367043 A357152 A146964 * A116881 A107089 A350480
KEYWORD
nonn,changed
AUTHOR
Vladimir Kruchinin, Aug 11 2011
STATUS
approved