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E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^4)/n!.
3

%I #7 Apr 07 2012 17:13:06

%S 1,15,4125,4201207,10454906015,51619504083157,445183896786430439,

%T 6151183312376366042809,127892318444027363237894001,

%U 3815107763405827557700743314007,157278812586433713743644391748289829,8693308684725580082237757157480179540583

%N E.g.f.: exp(-1)*Sum_{n>=0} (1+x)^(n^4)/n!.

%F a(n) = Sum_{k=0..n} Stirling1(n, k)*Bell(4*k).

%F a(n) = n!*exp(-1)*Sum_{k>=[n^(1/4)]} binomial(k^4,n)/k!.

%e E.g.f.: A(x) = 1 + 15*x + 4125*x^2/2! + 4201207*x^3/3! + 10454906015*x^4/4! +...

%e such that

%e A(x) = exp(-1)*(1 + (1+x) + (1+x)^16/2! + (1+x)^81/3! + (1+x)^256/4! +...).

%o (PARI) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

%o {Bell(n)=n!*polcoeff(exp(exp(x+x*O(x^n))-1), n)}

%o {a(n)=sum(k=0, n, Stirling1(n, k)*Bell(4*k))}

%o for(n=0,15,print1(a(n),", "))

%Y Cf. A000110 (Bell), A014507, A211250, A211252.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Apr 07 2012