A187803 - OEIS

%I #9 Nov 03 2014 15:47:16

%S 1,1,15,829,113487,31137061,15015039495,11636995485949,

%T 13584094722071007,22735343245138118101,52487807127760090483575,

%U 162018777092079952134169069,651747862300297714019151918927,3344015911143306355676226376118341,21488215819992049616143504500848490855

%N E.g.f.: Sum_{n>=0} Product_{k=1..n} (1 - exp(-n*k*x)).

%C Compare to the e.g.f. of A079144, enumerating certain labeled interval orders:

%C Sum_{n>=0} Product_{k=1..n} (1 - exp(-k*x)).

%C Also compare to the e.g.f. of A220181: Sum_{n>=0} (1 - exp(-n*x))^n.

%H Vaclav Kotesovec, <a href="/A187803/b187803.txt">Table of n, a(n) for n = 0..140</a>

%F a(n) ~ c * d^n * (n!)^3 / sqrt(n), where d = 2.426663845780394275167988381..., c = 0.504146101604802096078745... . - _Vaclav Kotesovec_, Nov 03 2014

%e E.g.f.: A(x) = 1 + x + 15*x^2/2! + 829*x^3/3! + 113487*x^4/4! +...

%e where

%e A(x) = 1 + (1-exp(-x)) + (1-exp(-2*1*x))*(1-exp(-2*2*x)) + (1-exp(-3*1*x))*(1-exp(-3*2*x))*(1-exp(-3*3*x)) + (1-exp(-4*1*x))*(1-exp(-4*2*x))*(1-exp(-4*3*x))*(1-exp(-4*4*x)) +...

%o (PARI) {a(n)=n!*polcoeff(sum(m=0, n, prod(k=1,m,(1-exp(-m*k*x+x*O(x^n)))) ), n)}

%o for(n=0, 20, print1(a(n), ", "))

%Y Cf. A079144, A220181.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jan 06 2013