%I #9 Jun 10 2019 21:20:34

%S 1,4,99,21924,45207549,864861114348,151334173143255375,

%T 240066309264838323117084,3437872832425973181485795041113,

%U 443629285029172409524181790790692095604,515464807018375729400140781858676274403447441691,5391365666991000164547212259503680126841305476860172028212

%N E.g.f.: Sum_{n>=0} 3^(n^2) * (1+x)^(3^n) * x^n / n!.

%C More generally, the following sums are equal:

%C (1) Sum_{n>=0} (q^n + p)^n * r^n / n!,

%C (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;

%C here, q = 3 and p = log(1+x)/x, r = x.

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

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

%F E.g.f.: Sum_{n>=0} 3^(n^2) * (1+x)^(3^n) * x^n / n!.

%F a(n) = n! * Sum_{k=0..n} 3^(k^2) * binomial(3^k,n-k) / k!.

%e E.g.f.: A(x) = 1 + 4*x + 99*x^2/2! + 21924*x^3/3! + 45207549*x^4/4! + 864861114348*x^5/5! + 151334173143255375*x^6/6! + 240066309264838323117084*x^7/7! + ...

%e such that

%e A(x) = 1 + 3*(1+x)^3*x + 3^4*(1+x)^9*x^2/2! + 3^9*(1+x)^27*x^3/3! + 3^16*(1+x)^81*x^4/4! + 3^25*(1+x)^243*x^5/5! + 3^36*(1+x)^729*x^6/6! + ...

%e also

%e A(x) = 1 + (3*x + log(1+x)) + (3^2*x + log(1+x))^2/2! + (3^3*x + log(1+x))^3/3! + (3^4*x + log(1+x))^4/4! + (3^5*x + log(1+x))^5/5! + (3^6*x + log(1+x))^6/6! + ...

%o (PARI) {a(n) = my(A = sum(m=0,n, 3^(m^2) * (1+x +x*O(x^n))^(3^m) * x^m/m!)); n!*polcoeff(A,n)}

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

%o (PARI) {a(n) = n! * sum(k=0, n, 3^(k^2) * binomial(3^k, n-k)/k!)}

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

%Y Cf. A326084.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 10 2019