A245060 a(n) = Sum_{k=1..n} C(n-1,k-1) * S2(n,k) * 3^(n-k) for n>0, a(0)=1, where S2(n,k) = A048993(n,k) are Stirling numbers of the 2nd kind.

1, 1, 4, 28, 271, 3172, 43174, 666577, 11445214, 215478712, 4401799930, 96757165012, 2273105615356, 56755763435503, 1499039156935948, 41714498328290992, 1218787798107634291, 37275555462806318512, 1190200470204107432854, 39581409916012393962280, 1368112674516484881342244

Offset

0,3

Links

Table of n, a(n) for n=0..20.

Formula

O.g.f.: Sum_{n>=0} (n*x)^n/(1-3*n*x)^n * exp(-n*x/(1-3*n*x)) / n!.

Example

O.g.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 271*x^4 + 3172*x^5 + 43174*x^6 +...
where
A(x) = 1 + x/(1-3*x)*exp(-x/(1-3*x)) + 2^2*x^2/(1-6*x)^2*exp(-2*x/(1-6*x))/2! + 3^3*x^3/(1-9*x)^3*exp(-3*x/(1-9*x))/3! + 4^4*x^4/(1-12*x)^4*exp(-4*x/(1-12*x))/4! +...
simplifies to a power series in x with integer coefficients.
Illustrate the definition of the terms by:
a(2) = 1*1*3 + 1*1 = 4;
a(3) = 1*1*3^2 + 2*3*3 + 1*1 = 28;
a(4) = 1*1*3^3 + 3*7*3^2 + 3*6*3 + 1*1 = 271;
a(5) = 1*1*3^4 + 4*15*3^3 + 6*25*3^2 + 4*10*3 + 1*1 = 3172;
a(6) = 1*1*3^5 + 5*31*3^4 + 10*90*3^3 + 10*65*3^2 + 5*15*3 + 1*1 = 43174; ...

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)*3^(n-k)))}
for(n=0, 25, print1(a(n), ", "))
(PARI) {a(n)=polcoeff(sum(k=0, n+1, (k*x)^k/(1-3*k*x)^k*exp(-k*x/(1-3*k*x+x*O(x^n)))/k!), n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A134055, A245059, A218667, A218670.

Sequence in context: A138272 A367470 A361049 * A191686 A231694 A112915

Adjacent sequences: A245057 A245058 A245059 * A245061 A245062 A245063

Keyword

nonn

Author

Paul D. Hanna, Jul 10 2014

Status

approved