A245059 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.

1, 1, 3, 17, 129, 1177, 12463, 149053, 1975473, 28628865, 449059179, 7562334793, 135837896769, 2588529249737, 52093016105575, 1102851978691749, 24480094135644513, 568066476383361793, 13745454515733689427, 346020796943921077057, 9043636093339718229697, 244954584886648170627641

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

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

Example

O.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 129*x^4 + 1177*x^5 + 12463*x^6 +...
where
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! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(2) = 1*1*2 + 1*1 = 3;
a(3) = 1*1*2^2 + 2*3*2 + 1*1 = 17;
a(4) = 1*1*2^3 + 3*7*2^2 + 3*6*2 + 1*1 = 129;
a(5) = 1*1*2^4 + 4*15*2^3 + 6*25*2^2 + 4*10*2 + 1*1 = 1177;
a(6) = 1*1*2^5 + 5*31*2^4 + 10*90*2^3 + 10*65*2^2 + 5*15*2 + 1*1 = 12463; ...

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)*2^(n-k)))}
for(n=0, 25, print1(a(n), ", "))
(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)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A134055, A245060, A218667, A218670.

Sequence in context: A187613 A179300 A199493 * A321587 A089815 A328402

Adjacent sequences: A245056 A245057 A245058 * A245060 A245061 A245062

Keyword

nonn

Author

Paul D. Hanna, Jul 10 2014

Status

approved