A277610 - OEIS

%I #17 Sep 22 2017 21:41:21

%S 1,1,5,36,339,3999,57388,977577,19348425,436886364,11084633347,

%T 312102694743,9653262860564,325242329821529,11853828646799153,

%U 464582888781914004,19481645509391087747,870252961810204549919,41253445365917239409916,2068244310629828065675481,109336176534540098236055769,6078206718063279979791668252,354471031348340363987467541507,21638266052947649126008431859703,1379839169160669434086676475756260

%N G.f.: 1 / (1 - Sum_{k>=1} k^k * x^k ).

%H Robert Israel, <a href="/A277610/b277610.txt">Table of n, a(n) for n = 0..386</a>

%F G.f.: exp( Sum_{n>=1} [ Sum_{k>=1} k^k * x^k ]^n / n ).

%F a(n) ~ n^n * (1 + 2*exp(-1)/n). - _Vaclav Kotesovec_, Nov 06 2016

%F "INVERT" transform of A000312. - _Alois P. Heinz_, Sep 22 2017

%e 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 +...

%e The logarithm of the g.f. begins:

%e 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 +...

%e which equals the sum

%e log(A(x)) = (x + 4*x^2 + 27*x^3 + 256*x^4  + 3125*x^5 +...) +

%e (x^2 + 8*x^3 + 70*x^4 + 728*x^5 + 9027*x^6 + 132136*x^7 +...)/2 +

%e (x^3 + 12*x^4 + 129*x^5 + 1480*x^6 + 19002*x^7 +...)/3 +

%e (x^4 + 16*x^5 + 204*x^6 + 2576*x^7 + 34602*x^8 +...)/4 +

%e (x^5 + 20*x^6 + 295*x^7 + 4080*x^8 + 57635*x^9 +...)/5 +

%e (x^6 + 24*x^7 + 402*x^8 + 6056*x^9 + 90165*x^10 +...)/6 +

%e (x^7 + 28*x^8 + 525*x^9 + 8568*x^10 + 134512*x^11 +...)/7 +

%e ... +

%e (x + 2^2*x^2 + 3^3*x^3 + 4^4*x^4 + 5^5*x^5 +...+ k^k*x^k +...)^n/n +

%e ...

%p G:= 1/(1-Sum(k^k*x^k,k=1..infinity)):

%p S:= series(G,x,51):

%p seq(coeff(S,x,j),j=0..50); # _Robert Israel_, Nov 06 2016

%p # second Maple program:

%p a:= proc(n) option remember; `if`(n=0, 1,

%p       add(j^j*a(n-j), j=1..n))

%p     end:

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 22 2017

%t CoefficientList[Series[1/(1 - Sum[k^k * x^k, {k, 1, 20}]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Nov 06 2016 *)

%o (PARI) {a(n) = polcoeff( 1/(1 - sum(k=1, n+1, k^k * x^k +x*O(x^n)) ), n)}

%o for(n=0, 30, print1(a(n), ", "))

%Y Cf. A000312, A088342, A277611.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Oct 23 2016