OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..518
FORMULA
O.g.f.: Sum_{n>=0} (n*x)^n/(1-n*x)^n * exp(-n*x/(1-n*x)) / n!. - Paul D. Hanna, Nov 04 2012
From Alois P. Heinz, Jun 24 2023: (Start)
a(n) mod 2 = A037011(n) for n >= 1.
a(n) mod 2 = 1 <=> n in { A048297 } or n = 0. (End)
EXAMPLE
O.g.f.: A(x) = 1 + x + 2*x^2 + 8*x^3 + 41*x^4 + 252*x^5 + 1782*x^6 + 14121*x^7 +...
where
A(x) = 1 + x/(1-x)*exp(-x/(1-x)) + 2^2*x^2/(1-2*x)^2*exp(-2*x/(1-2*x))/2! + 3^3*x^3/(1-3*x)^3*exp(-3*x/(1-3*x))/3! + 4^4*x^4/(1-4*x)^4*exp(-4*x/(1-4*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(4) = 1*1 + 3*7 + 3*6 + 1*1 = 41;
a(5) = 1*1 + 4*15 + 6*25 + 4*10 + 1*1 = 252;
a(6) = 1*1 + 5*31 + 10*90 + 10*65 + 5*15 + 1*1 = 1782.
MAPLE
a:= proc(n) option remember; local b; b:=
proc(h, m) option remember; `if`(h=0,
binomial(n-1, m-1), m*b(h-1, m)+b(h-1, m+1) )
end; b(n, 0)
end:
seq(a(n), n=0..22); # Alois P. Heinz, Jun 24 2023
MATHEMATICA
Flatten[{1, Table[Sum[Binomial[n-1, k-1] * StirlingS2[n, k], {k, 1, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Aug 11 2014 *)
PROG
(PARI) a(n)=if(n==0, 1, sum(k=1, n, binomial(n-1, k-1)*polcoeff(1/prod(i=0, k, 1-i*x +x*O(x^(n-k))), n-k)))
(PARI) a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-k*x)^k*exp(-k*x/(1-k*x+x*O(x^n)))/k!), n)
for(n=0, 25, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 04 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 08 2007
EXTENSIONS
An initial '1' was added and definition changed slightly by Paul D. Hanna, Nov 04 2012
STATUS
approved