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a(n) = Sum_{k=0..n} binomial(n,k)*3^k*(k+1)^(n-k).
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%I #13 Jan 20 2017 18:31:39

%S 1,4,22,145,1096,9259,85924,865183,9364864,108173827,1325589676,

%T 17149360111,233271228880,3324545097475,49493784653644,

%U 767665750130839,12376226335249024,206967901014192643,3583561993192959436,64136093489935863583,1184711492540805987856

%N a(n) = Sum_{k=0..n} binomial(n,k)*3^k*(k+1)^(n-k).

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

%F E.g.f.: exp(x + 3*x*exp(x)).

%t Table[Sum[Binomial[n,k]3^k (k+1)^(n-k),{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Nov 12 2012 *)

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*3^k*(k+1)^(n-k))}

%o (PARI) {a(n)=polcoeff(sum(m=0,n,3^m*x^m/(1-(m+1)*x+x*O(x^n))^(m+1)),n)}

%o (PARI) {a(n)=n!*polcoeff(exp(x+3*x*exp(x+x*O(x^n))),n)}

%Y Cf. A080108, A196794.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Oct 06 2011