OFFSET
0,3
LINKS
Robert Israel, Table of n, a(n) for n = 0..386
FORMULA
G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^k * x^k ]^n / n ).
a(n) ~ n^n * (1 + 2*exp(-1)/n). - Vaclav Kotesovec, Nov 06 2016
"INVERT" transform of A000312. - Alois P. Heinz, Sep 22 2017
EXAMPLE
G.f.: A(x) = 1 + x + 5*x^2 + 36*x^3 + 339*x^4 + 3999*x^5 + 57388*x^6 + 977577*x^7 + 19348425*x^8 + 436886364*x^9 + 11084633347*x^10 +...
The logarithm of the g.f. begins:
log(A(x)) = x + 9*x^2/2 + 94*x^3/3 + 1181*x^4/4 + 17681*x^5/5 + 310308*x^6/6 + 6276565*x^7/7 + 144052445*x^8/8 + 3701113150*x^9/9 + 105252411369*x^10/10 + 3281812541569*x^11/11 + 111313031195216*x^12/12 + 4079782609460013*x^13/13 + 160665945152295921*x^14/14 + 6765274535733165854*x^15/15 +...
which equals the sum
log(A(x)) = (x + 4*x^2 + 27*x^3 + 256*x^4 + 3125*x^5 +...) +
(x^2 + 8*x^3 + 70*x^4 + 728*x^5 + 9027*x^6 + 132136*x^7 +...)/2 +
(x^3 + 12*x^4 + 129*x^5 + 1480*x^6 + 19002*x^7 +...)/3 +
(x^4 + 16*x^5 + 204*x^6 + 2576*x^7 + 34602*x^8 +...)/4 +
(x^5 + 20*x^6 + 295*x^7 + 4080*x^8 + 57635*x^9 +...)/5 +
(x^6 + 24*x^7 + 402*x^8 + 6056*x^9 + 90165*x^10 +...)/6 +
(x^7 + 28*x^8 + 525*x^9 + 8568*x^10 + 134512*x^11 +...)/7 +
... +
(x + 2^2*x^2 + 3^3*x^3 + 4^4*x^4 + 5^5*x^5 +...+ k^k*x^k +...)^n/n +
...
MAPLE
G:= 1/(1-Sum(k^k*x^k, k=1..infinity)):
S:= series(G, x, 51):
seq(coeff(S, x, j), j=0..50); # Robert Israel, Nov 06 2016
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1,
add(j^j*a(n-j), j=1..n))
end:
seq(a(n), n=0..20); # Alois P. Heinz, Sep 22 2017
MATHEMATICA
CoefficientList[Series[1/(1 - Sum[k^k * x^k, {k, 1, 20}]), {x, 0, 20}], x] (* Vaclav Kotesovec, Nov 06 2016 *)
PROG
(PARI) {a(n) = polcoeff( 1/(1 - sum(k=1, n+1, k^k * x^k +x*O(x^n)) ), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 23 2016
STATUS
approved