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%I #17 Jul 06 2019 09:21:38
%S 1,3,39,948,34869,1757163,114320118,9226773993,897658726215,
%T 103005144933870,13705015429716807,2085418048857405375,
%U 358813807291982519184,69146346672687725451039,14803634157756603606592167,3496440993213535696041009924,905508769623362527769907535857,255762146004426658313683324505247,78413243604482526944814375388910526,25983968388767783226046397856822603645
%N E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} (p + q^n)^n * r^n/n!,
%C (2) Sum_{n>=0} q^(n^2) * exp(p*q^n*r) * r^n/n!;
%C here, q = exp(x) with p = -1, r = 3.
%C In general, let F(x) be a formal power series in x such that F(0)=1, then
%C Sum_{n>=0} m^n * F(q^n*r)^b * log( F(q^n*r) )^n / n! =
%C Sum_{n>=0} r^n * [y^n] F(y)^(m*q^n + p);
%C here, F(x) = exp(x), q = exp(x), p = -1, r = 3, m = 1.
%H Paul D. Hanna, <a href="/A326271/b326271.txt">Table of n, a(n) for n = 0..300</a>
%F E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n!.
%F E.g.f.: Sum_{n>=0} 3^n * exp(n^2*x) * exp( -3*exp(n*x) ) / n!.
%F O.g.f.: Sum_{n>=0} 3^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x).
%F a(n) = Sum_{k=0..n} 3^k * k^n * Stirling2(n,k).
%e E.g.f.: A(x) = 1 + 3*x + 39*x^2/2! + 948*x^3/3! + 34869*x^4/4! + 1757163*x^5/5! + 114320118*x^6/6! + 9226773993*x^7/7! + 897658726215*x^8/8! + 103005144933870*x^9/9! + ...
%e such that
%e A(x) = 1 + 3*(exp(x) - 1) + 3^2*(exp(2*x) - 1)^2/2! + 3^3*(exp(3*x) - 1)^3/3! + 3^4*(exp(4*x) - 1)^4/4! + 3^5*(exp(5*x) - 1)^5/5! + 3^6*(exp(6*x) - 1)^6/6! + ...
%e also
%e A(x) = exp(-3) + 3*exp(x)*exp(-3*exp(x)) + 3^2*exp(4*x)*exp(-3*exp(2*x))/2! + 3^3*exp(9*x)*exp(-3*exp(3*x))/3! + 3^4*exp(16*x)*exp(-3*exp(4*x))/4! + 3^5*exp(25*x)*exp(-3*exp(5*x))/5! + 3^6*exp(36*x)*exp(-3*exp(6*x))/6! + ...
%e ORDINARY GENERATING FUNCTION.
%e O.g.f.: B(x) = 1 + 3*x + 39*x^2 + 948*x^3 + 34869*x^4 + 1757163*x^5 + 114320118*x^6 + 9226773993*x^7 + 897658726215*x^8 + 103005144933870*x^9 + ...
%e such that
%e B(x) = 1 + 3*x/(1-x) + 3^2*2^2*x^2/((1-2*x)*(1-4*x)) + 3^3*3^3*x^3/((1-3*x)*(1-6*x)*(1-9*x)) + 3^4*4^4*x^4/((1-4*x)*(1-8*x)*(1-12*x)*(1-16*x)) + 3^5*5^5*x^5/((1-5*x)*(1-10*x)*(1-15*x)*(1-20*x)*(1-25*x)) + ...
%o (PARI) {a(n) = sum(k=0, n, 3^k * k^n * stirling(n, k, 2) )}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* E.g.f.: Sum_{n>=0} 3^n * (exp(n*x) - 1)^n / n! */
%o {a(n) = n! * polcoeff(sum(m=0, n, 3^m * (exp(m*x +x*O(x^n)) - 1)^m / m!), n)}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) /* O.g.f.: Sum_{n>=0} 3^n * n^n * x^n / Product_{k=1..n} (1 - n*k*x) */
%o {a(n) = polcoeff(sum(m=0, n, 3^m * m^m * x^m / prod(k=1, m, 1-m*k*x +x*O(x^n))), n)}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A108459, A326270, A326288.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 28 2019
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