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%I #19 Jul 06 2022 05:39:45
%S 1,3,15,90,633,5028,44217,424434,4399953,48858984,577372809,
%T 7221983838,95192539641,1317190650636,19071213218745,288112248054882,
%U 4530217559806497,73976635012027344,1252091246140278153,21926952634345281030,396671314081806278601
%N Expansion of e.g.f. exp(3 * x * exp(x)).
%F G.f.: Sum_{k>=0} (3 * x)^k/(1 - k*x)^(k+1).
%F a(0) = 1; a(n) = 3 * Sum_{k=1..n} k * binomial(n-1,k-1) * a(n-k).
%F a(n) = Sum_{k=0..n} 3^k * k^(n-k) * binomial(n,k).
%F a(n) ~ n^(n + 1/2) * exp(3*r*exp(r) - r/2 - n) / (sqrt(3*(1 + 3*r + r^2)) * r^(n + 1/2)), where r = 2*w - 1/(2*w) + 5/(8*w^2) - 19/(24*w^3) + 209/(192*w^4) - 763/(480*w^5) + 4657/(1920*w^6) - 6855/(1792*w^7) + 199613/(32256*w^8) + ... and w = LambertW(sqrt(n/3)/2). - _Vaclav Kotesovec_, Jul 06 2022
%o (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*x*exp(x))))
%o (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (3*x)^k/(1-k*x)^(k+1)))
%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=3*sum(j=1, i, j*binomial(i-1, j-1)*v[i-j+1])); v;
%o (PARI) a(n) = sum(k=0, n, 3^k*k^(n-k)*binomial(n, k));
%Y Column 3 of A187105.
%Y Cf. A000248, A275707, A295623.
%Y Cf. A351763.
%K nonn
%O 0,2
%A _Seiichi Manyama_, Jul 04 2022