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

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

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

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

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

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

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

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allocated for Paul D. Hanna

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

1, 4, 99, 21924, 45207549, 864861114348, 151334173143255375, 240066309264838323117084, 3437872832425973181485795041113, 443629285029172409524181790790692095604, 515464807018375729400140781858676274403447441691, 5391365666991000164547212259503680126841305476860172028212

0,2

More generally, the following sums are equal:

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

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

here, q = 3*x and p = log(1+x)/x.

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

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

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! + ...

such that

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! + ...

also

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! + ...

(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)}

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

Cf. A326084.

allocated

nonn

Paul D. Hanna, Jun 10 2019

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allocated for Paul D. Hanna

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