A245059 - OEIS

%I #6 Jul 10 2014 20:40:07

%S 1,1,3,17,129,1177,12463,149053,1975473,28628865,449059179,7562334793,

%T 135837896769,2588529249737,52093016105575,1102851978691749,

%U 24480094135644513,568066476383361793,13745454515733689427,346020796943921077057,9043636093339718229697,244954584886648170627641

%N a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 2^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

%F O.g.f.: Sum_{n>=0} (n*x)^n/(1-2*n*x)^n * exp(-n*x/(1-2*n*x)) / n!.

%e O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 129*x^4 + 1177*x^5 + 12463*x^6 +...

%e where

%e A(x) = 1 + x/(1-2*x)*exp(-x/(1-2*x)) + 2^2*x^2/(1-4*x)^2*exp(-2*x/(1-4*x))/2! + 3^3*x^3/(1-6*x)^3*exp(-3*x/(1-6*x))/3! + 4^4*x^4/(1-8*x)^4*exp(-4*x/(1-8*x))/4! +...

%e simplifies to a power series in x with integer coefficients.

%e Illustrate the definition of the terms by:

%e a(2) = 1*1*2 + 1*1 = 3;

%e a(3) = 1*1*2^2 + 2*3*2 + 1*1 = 17;

%e a(4) = 1*1*2^3 + 3*7*2^2 + 3*6*2 + 1*1 = 129;

%e a(5) = 1*1*2^4 + 4*15*2^3 + 6*25*2^2 + 4*10*2 + 1*1 = 1177;

%e a(6) = 1*1*2^5 + 5*31*2^4 + 10*90*2^3 + 10*65*2^2 + 5*15*2 + 1*1 = 12463; ...

%o (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)*2^(n-k)))}

%o for(n=0, 25, print1(a(n), ", "))

%o (PARI) {a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-2*k*x)^k*exp(-k*x/(1-2*k*x+x*O(x^n)))/k!), n)}

%o for(n=0, 25, print1(a(n), ", "))

%Y Cf. A134055, A245060, A218667, A218670.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jul 10 2014