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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

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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), ", "))