cp's OEIS Frontend

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.

%I A245060 #5 Jul 10 2014 20:40:26
%S A245060 1,1,4,28,271,3172,43174,666577,11445214,215478712,4401799930,
%T A245060 96757165012,2273105615356,56755763435503,1499039156935948,
%U A245060 41714498328290992,1218787798107634291,37275555462806318512,1190200470204107432854,39581409916012393962280,1368112674516484881342244
%N 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.
%F A245060 O.g.f.: Sum_{n>=0} (n*x)^n/(1-3*n*x)^n * exp(-n*x/(1-3*n*x)) / n!.
%e A245060 O.g.f.: A(x) = 1 + x + 4*x^2 + 28*x^3 + 271*x^4 + 3172*x^5 + 43174*x^6 +...
%e A245060 where
%e A245060 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! +...
%e A245060 simplifies to a power series in x with integer coefficients.
%e A245060 Illustrate the definition of the terms by:
%e A245060 a(2) = 1*1*3 + 1*1 = 4;
%e A245060 a(3) = 1*1*3^2 + 2*3*3 + 1*1 = 28;
%e A245060 a(4) = 1*1*3^3 + 3*7*3^2 + 3*6*3 + 1*1 = 271;
%e A245060 a(5) = 1*1*3^4 + 4*15*3^3 + 6*25*3^2 + 4*10*3 + 1*1 = 3172;
%e A245060 a(6) = 1*1*3^5 + 5*31*3^4 + 10*90*3^3 + 10*65*3^2 + 5*15*3 + 1*1 = 43174; ...
%o A245060 (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)))}
%o A245060 for(n=0, 25, print1(a(n), ", "))
%o A245060 (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)}
%o A245060 for(n=0, 25, print1(a(n), ", "))
%Y A245060 Cf. A134055, A245059, A218667, A218670.
%K A245060 nonn
%O A245060 0,3
%A A245060 _Paul D. Hanna_, Jul 10 2014