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%I #9 Jul 21 2023 15:39:33
%S 1,4,100,10648,4477456,7339040224,47045881000000,1186980379913527168,
%T 118530511097526559703296,47035767668340696232372862464,
%U 74367598058372171073462490000000000,469253945833810205185008441288962454059008
%N E.g.f: Sum_{n>=0} 3^n * 2^(n^2) * exp(-2*2^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*2^n - 2)^n.
%F O.g.f.: Sum_{n>=0} 3^n * 2^(n^2) * x^n/(1 + 2*2^n*x)^(n+1).
%e E.g.f.: A(x) = 1 + 4*x + 100*x^2/2! + 10648*x^3/3! + 4477456*x^4/4! +..
%e By the series identity, the e.g.f.:
%e A(x) = exp(-2*x) + 3*2*exp(-2*2*x)*x + 3^2*2^4*exp(-2*2^2*x)*x^2/2! + 3^3*2^9*exp(-2*2^3*x)*x^3/3! +...
%e expands into:
%e A(x) = 1 + 4*x + 10^2*x^2/2! + 22^3*x^3/3! + 46^4*x^4/4! + 94^5*x^5/5! +...+ (3*2^n-2)^n*x^n/n! +...
%t Table[(3*2^n-2)^n,{n,0,12}] (* _Harvey P. Dale_, Jul 16 2023 *)
%o (PARI) {a(n, q=2, m=3, b=-2)=(m*q^n + b)^n}
%o (PARI) {a(n, q=2, m=3, b=-2)=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=2, m=3, b=-2)=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. A180602, A165327, A202989, A060613, A055601.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 26 2011
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