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A202989 E.g.f: Sum_{n>=0} 3^(n^2) * exp(3^n*x) * x^n/n!. 6

%I

%S 1,4,100,21952,45212176,864866612224,151334226289000000,

%T 240066313618039143841792,3437872835498096514323500400896,

%U 443629285048033016198674962874808664064,515464807019389919369209932597753906250000000000

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

%C E.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * exp(b*q^n*x) * x^n/n! = Sum_{n>=0} (m*q^n + b)^n * x^n/n! for all q, m, b.

%C O.g.f. series identity: Sum_{n>=0} m^n * q^(n^2) * x^n/(1-b*q^n*x)^(n+1) = Sum_{n>=0} (m*q^n + b)^n * x^n for all q, m, b.

%F a(n) = (3^n + 1)^n.

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

%e E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 21952*x^3/3! + 45212176*x^4/4! +..

%e By the series identity, the e.g.f.:

%e A(x) = exp(x) + 3*exp(3*x)*x + 3^4*exp(3^2*x)*x^2/2! + 3^9*exp(3^3*x)*x^3/3! +...

%e expands into:

%e A(x) = 1 + 4*x + 10^2*x^2/2! + 28^3*x^3/3! + 82^4*x^4/4! + 244^5*x^5/5! +...+ (3^n+1)^n*x^n/n! +...

%o (PARI) {a(n, q=3, m=1, b=1)=(m*q^n + b)^n}

%o (PARI) {a(n, q=3, m=1, b=1)=n!*polcoeff(sum(k=0, n, m^k*q^(k^2)*exp(b*q^k*x+x*O(x^n))*x^k/k!), n)}

%o (PARI) {a(n, q=3, m=1, b=1)=polcoeff(sum(k=0,n,m^k*q^(k^2)*x^k/(1-b*q^k*x +x*O(x^n))^(k+1)),n)}

%Y Cf. A165327, A060613, A244004, A136516, A326013.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Dec 26 2011

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Last modified April 2 18:43 EDT 2020. Contains 333189 sequences. (Running on oeis4.)